Defining New Structures on a Universal Set: Diving Structures and Floating Structures

Abstract

This paper introduces new structures ($\mathcal{DV}$, $\mathcal{FL}$) on a set $\mathrm{\Xi }$. They will be known as diving structures and floating structures. The pairs $(\mathrm{\Xi },\mathcal{DV})$, $(\mathrm{\Xi },\mathcal{FL})$ will be called diving spaces and floating spaces, respectively. These structures are not related to each other and are not related to any of the previous common structures such as topology, ideal, filter, grill or primal. We study some properties to characterize these new notions. Separation axioms and connectedness will be defined in these new spaces. When $\mathbb{R}$ is an equivalence relation on $\mathrm{\Xi }$, rough sets in diving spaces and rough sets in floating spaces will also be defined. Thus, $(\mathrm{\Xi },\mathbb{R},\mathcal{DV})$ and $(\mathrm{\Xi },\mathbb{R},\mathcal{FL})$ are called the diving approximation spaces and the floating approximation spaces, respectively.

1. Introduction

The motivation of this paper is to build two structures on a set $\mathrm{\Xi }$ according to the size of their elements. The first structure $\mathcal{DV}\subseteq P\left(\mathrm{\Xi }\right)$ is given as follows. The largest (heaviest) member must be found, and the members with a smaller size are found until a moment when the size is considered small enough (light). So, this structure represents the case of diving, where the largest member dives to the bottom, and the fewer members dive to the bottom until the members could be described light in relation to the already diving members. This builds a diving structure of specific subsets of a universal set $\mathrm{\Xi }$, and the pair $(\mathrm{\Xi },\mathcal{DV})$ is said to be a diving space. The second structure $\mathcal{FL}\subseteq P\left(\mathrm{\Xi }\right)$ is given as follows; the smallest (lightest) member must be found, and the members with greater size are found until the moment that the size is considered large enough (heavy). So, this structure represents the case of floating, where the smallest member floats to shore, and the greater members are floating to shore until they become members that could be described heavy in relation to the already floating members. This builds a floating structure of specific subsets of a universal set $\mathrm{\Xi }$, and the pair $(\mathrm{\Xi },\mathcal{FL})$ is said to be a floating space.

The motivation behind these structures stems from the need to establish alternative frameworks for understanding subsets within a power set. By extending the fundamental properties of union, intersection, inclusion, and complement, diving structures and floating structures allow for a more rich analysis of relationships between subsets. The significance of these structures lies in their potential applications across various disciplines, including rough set theory, data analysis, fuzzy logic, information system, and physics. Their ability to generalize and expand upon existing mathematical concepts position them as a foundational tool for future studies in abstract mathematics and applied sciences. This paper explores the properties and characteristics of these structures, defining key axioms and separation criteria that further distinguish them. For instance, diving and floating spaces introduce new forms of connectedness and disconnectedness, allowing for a reevaluation of traditional continuity and neighborhood concepts. Following [1], we define the roughness notion in both of these new spaces. Additionally, the extension of these structures to fuzzy sets suggests potential applications in systems requiring probabilistic or graded classifications, such as artificial intelligence and decision-making frameworks. Through detailed examples, the paper illustrates the practicality and adaptability of these structures, emphasizing their relevance to both theoretical and applied fields. Diving and floating structures represent a pioneering approach to set theory and topology, offering novel insights and tools for researchers. By transcending traditional limitations and introducing fresh perspectives, these structures have the potential to revolutionize the understanding and application of mathematical spaces.

The concepts of diving structures ($\mathcal{DV}$) and floating structures ($\mathcal{FL}$), as introduced in this research, represent a significant advancement in the study of set theory and topology. These novel structures redefine how subsets within a universal set are categorized and analyzed based on their intrinsic properties, particularly their “weight”. Diving structures are characterized by subsets descending hierarchically, akin to the “heaviest” subsets sinking to the bottom, while floating structures emphasize an ascending order, with the “lightest” subsets floating to the top. These structures operate independently of classical frameworks such as topologies, supra topologies, infra topologies, anti-topologies, filters, ideals, grills, primals, and generalized structures, creating unique opportunities for exploration. Diving structures will absorb some general concepts and give up some others, and oppositely floating structures will do. As examples, based on the nature of the diving subsets, the diving spaces yield to be connected more than to be disconnected, and, moreover, separation axioms are closer to being not satisfied than to being satisfied. Conversely, based on the nature of the floating subsets, the floating spaces yield to be disconnected more than to be connected, and, moreover, separation axioms are closer to being satisfied than to not being satisfied. Studying roughness in these new spaces will be completely different from the usual roughness in set theory.

According to the special conditions of the diving structures and the floating structures, which are different from those of any well known classical branch in set theory, the resulting diving spaces and floating spaces will be new branches in Pure Mathematics. We then need to examine or study lots of theoretical concepts in these spaces, and it will be ensured that the proposed studies will deviate from the well know studies of topologies, filters, ideals, grills, primals, or any well known classical branch in set theory.

Throughout the paper, let $\mathrm{\Xi }$ be a universal set of elements and I the closed unit interval $[0,1]$. Denote the interval $(0,1]$ by $I_0$ and denote the interval $[0,1)$ by $I_1$. Let $I^X$ denote all these fuzzy subsets [2] of $\mathrm{\Xi }$. A constant fuzzy set $\overline{r}$ for all $r\in I$ is defined by $\overline{r}\left(x\right)=r\forall x\in \mathrm{\Xi }$. A fuzzy point $x_t\in I^{\mathrm{\Xi }}$ for some $t\in I$ is defined by $x_t\left(y\right)=0\forall y\ne x$ and $x_t\left(x\right)=t$. A fuzzy point $x_t$ belongs to a fuzzy set U if $U\left(x\right)\ge t$, and $x_t$ does not belong to U if $U\left(x\right)<t$. $P\left(\mathrm{\Xi }\right)$ denotes the power set of $\mathrm{\Xi }$ and let $2^{\mathrm{\Xi }}$ be the set of all these characteristic functions on $\mathrm{\Xi }$. Meanwhile, it is well-known that in set theory, there is a bijection between $P\left(\mathrm{\Xi }\right)$ and $2^{\mathrm{\Xi }}$. So, writing $\mathbb{A}\in 2^{\mathrm{\Xi }}$ is equivalent to $\mathbb{A}\subseteq \mathrm{\Xi }$ without any distinction. Any structure S defined on $P\left(\mathrm{\Xi }\right)$ with the order relation ⊆ could be written in the mapping form ($S:2^{\mathrm{\Xi }}\to \{0,1\}$) given on $2^{\mathrm{\Xi }}$ with the order relation ≤. That mapping form is known by the crisp case representing the classical case as a special case of the fuzzy case.

In Section 2, we present all the defined structures on a set $\mathrm{\Xi }$, starting from the definition of general topology. All structures are built on reproducing conditions on the subsets taken from the universe $\mathrm{\Xi }$. All these previous structures will be summarized in an ascending manner. At the end, we introduce the new structures, diving structure and floating structure which are independent of all those previous structures in the summary. In Section 3, as one of the main applications in general topology, we define separation axioms in both diving spaces and floating spaces. We give some properties and examples for the applications of these new structures. In Section 4, we objected to another well-known application in topological spaces; it is the notion of connectedness and the notion of disconnectedness. An example showing the behavior of the connectedness in these new spaces is given. In Section 5, we define the notion of roughness in these new structures. Many properties and results are given. The common notion of roughness initiated by Pawlak in [3] is a particular case of the given in this section. Some examples are given to show the behavior of roughness in these new spaces. In Section 6, we define these structures in the fuzzy case. We simply show the possibility of defining these structures on fuzzy sets, and a detailed study will be produced in future. Section 7 will conclude the paper.

2. Historic Summary

In this section, we summarize most of the previous notions constructed on a set $\mathrm{\Xi }$ as a subset of the power set $P\left(\mathrm{\Xi }\right)$. At the end, we define the main structures we introduced in this paper, and study some of its properties. This research paper discusses two novel mathematical structures called diving structures ($\mathcal{DV}$) and floating structures ($\mathcal{FL}$), and their spaces, diving spaces and floating spaces, respectively. These are not related to any of the conventional topological and generalized structures.

Topological spaces were first defined by Felix Hausdorff in 1914 in his seminal Principles of Set Theory. A non-empty family $\tau$ of subsets of a non-empty set $\mathrm{\Xi }$ is called a topology on $\mathrm{\Xi }$ if it satisfies the following conditions:

(T1)

$\mathrm{\Xi },\varphi \in \tau$;

(T2)

If $\{{\mathbb{A}}_i:i\in J\}\subseteq \tau$, J a finite index set, then ${\displaystyle \bigcap _{i\in J}}{\mathbb{A}}_i\in \tau$;

(T3)

If $\{{\mathbb{A}}_i:i\in J\}\subseteq \tau$, J an arbitrary index set, then ${\displaystyle \bigcup _{i\in J}}{\mathbb{A}}_i\in \tau$.

The pair $(\mathrm{\Xi },\tau )$ is said to be a topological space.

Alexandroff in 1937 introduced the notion “Diskrete Räume”, which is known as Alexandroff spaces [4], by neglecting all the conditions of a topology and using only one condition: $\tau$ is closed under arbitrary intersections If $\{{\mathbb{A}}_i:i\in J\}\subseteq \tau$ and J an arbitrary index set, then ${\displaystyle \bigcap _{i\in J}}{\mathbb{A}}_i\in \tau$.

In 1937, Henri Cartan, initiated the definition of a filter on a set $\mathrm{\Xi }$ as follows: A non-empty family $\mathcal{F}$ of subsets of a non-empty set $\mathrm{\Xi }$ is called a filter [5] on $\mathrm{\Xi }$ if it satisfies the following conditions:

(F1)

$\varphi \notin \mathcal{F}$;

(F2)

If $\mathbb{A}\in \mathbb{B}$ and $\mathbb{A}\in \mathcal{F}$, then $\mathbb{B}\in \mathcal{F}$ for all $\mathbb{A},\mathbb{B}\in 2^{\mathrm{\Xi }}$;

(F3)

If $\mathbb{A}\in \mathcal{F}$ and $\mathbb{B}\in \mathcal{F}$, then $(\mathbb{A}\cap \mathbb{B})\in \mathcal{F}$ for all $\mathbb{A},\mathbb{B}\in 2^{\mathrm{\Xi }}$.

In 1947, Choquet, initiated the definition of a grill on a set $\mathrm{\Xi }$ as follows: A non-empty family $\mathcal{G}$ of subsets of a non-empty set $\mathrm{\Xi }$ is called a grill [6] on $\mathrm{\Xi }$ if it satisfies the conditions (F1) and (F2) of a filter in addition to a different condition than (F3). This means that $\mathcal{G}$ satisfies the following conditions:

(G1)

$\varphi \notin \mathcal{G}$;

(G2)

If $\mathbb{A}\subseteq \mathbb{B} \mathrm{and} \mathbb{A}\in \mathcal{G}$, then $\mathbb{B}\in \mathcal{G}$ for all $\mathbb{A},\mathbb{B}\in 2^{\mathrm{\Xi }}$;

(G3)

If $(\mathbb{A}\cup \mathbb{B})\in \mathcal{G}$, then $\mathbb{A}\in \mathcal{G}$ or $\mathbb{B}\in \mathcal{G}$ for all $\mathbb{A},\mathbb{B}\in 2^{\mathrm{\Xi }}$.

The notion of an ideal on $\mathrm{\Xi }$ is given by Kuratowski in 1966 as follows: A non-empty family $\mathcal{I}$ of subsets of a non-empty set $\mathrm{\Xi }$ is called an ideal [7] on $\mathrm{\Xi }$ if it satisfies the following conditions:

(I1)

$\mathrm{\Xi }\notin \mathcal{I}$;

(I2)

If $\mathbb{A}\subseteq \mathbb{B} \mathrm{and} \mathbb{B}\in \mathcal{I}$, then $\mathbb{A}\in \mathcal{I}$ for all $\mathbb{A},\mathbb{B}\in 2^{\mathrm{\Xi }}$;

(I3)

If $\mathbb{A}\in \mathcal{I}$ and $\mathbb{B}\in \mathcal{I}$, then $(\mathbb{A}\cup \mathbb{B})\in \mathcal{I}$ for all $\mathbb{A},\mathbb{B}\in 2^{\mathrm{\Xi }}$.

In 1966, it was given the original definition of generalized weak structures by Kim-Leong Lim in [8]. However, Lim used the notions “Abstract spaces” or “Generalized topological spaces”. Lim supposed that S is a structure on $\mathrm{\Xi }$ without any of the conditions of a topology. Later, in 2012, Avila and Molina [9] defined the same structure as Lim with the new title of “Generalized weak structures”. So, the results in [9] are not new, it is only that they began the notation “Generalized weak structures”. The pair $(\mathrm{\Xi },S)$ is said to be a generalized weak space.

Mashhour et al., in 1983, introduced the supra topological spaces by neglecting the condition (T2) of a topology: that is, $\tau$ is a supra topology [10] on $\mathrm{\Xi }$ if $\left(T1\right)$ and $\left(T3\right)$ are satisfied.

In 1996, Maki introduced the notion of a minimal structure [11] $m_X$ on a set $\mathrm{\Xi }$ by the assumption that only the condition (T1) of a topology is satisfied. The pair $(\mathrm{\Xi },m_X)$ is called a minimal space.

Császár, in 2002, introduced the notion of a generalized topology [12] G on $\mathrm{\Xi }$ as follows: G satisfies that $\varphi \in G$ and the condition (T3) of a topology is fulfilled (that means generalized topology G with $\mathrm{\Xi }\in G$, implying that G is a supra topology). The pair $(\mathrm{\Xi },G)$ is said to be a generalized topological space.

Also, Császár, in 2011, introduced the notion of a weak structure [13] w on $\mathrm{\Xi }$ as follows: w satisfies only that $\varphi \in w$ (that means a weak structure w with (T3) of a topology is fulfilled, implying that w is a generalized topology). The pair $(\mathrm{\Xi },w)$ is called a weak topological space.

In 2015, Al-Odhari introduced the notion of infra topological spaces by neglecting the condition (T3) of a topology: that is, ${\tau }^{*}$ is an infra topology [14] on $\mathrm{\Xi }$ if $\left(T1\right)$ and $\left(T2\right)$ are satisfied. The pair $(\mathrm{\Xi },{\tau }^{*})$ is called an infra topological space.

Sahin et al., in 2021, defined the notion of anti-topological spaces as follows: A non-empty family $\mathcal{A}$ of subsets of a non-empty set $\mathrm{\Xi }$ is called an anti-topology [15] on $\mathrm{\Xi }$ if it satisfies the following conditions:

(A1)

$\varphi ,\mathrm{\Xi }\notin \mathcal{A}$;

(A2)

If $\{{\mathbb{B}}_i;i\in J\}\subseteq \mathcal{A}$, J a finite index set, then ${\displaystyle \bigcap _{i\in J}}{\mathbb{B}}_i\notin \mathcal{A}$;

(A3)

If $\{{\mathbb{B}}_i;i\in J\}\subseteq \mathcal{A}$, J an arbitrary index set, then ${\displaystyle \bigcup _{i\in J}}{\mathbb{B}}_i\notin \mathcal{A}$.

The pair $(\mathrm{\Xi },\mathcal{A})$ is called an anti-topological space.

In 2022, Acharjee et al. produced the definition of a primal on a set $\mathrm{\Xi }$ as follows: A non-empty family $\mathcal{P}$ of subsets of a non-empty set $\mathrm{\Xi }$ is said to be a primal [16] on $\mathrm{\Xi }$ if it satisfies the conditions (I1) and (I2) of an ideal in addition to a different condition than (I3), that is, if $\mathcal{P}$ satisfies the following conditions:

(P1)

$\mathrm{\Xi }\notin \mathcal{P}$;

(P2)

If $\mathbb{A}\subseteq \mathbb{B} \mathrm{and} \mathbb{B}\in \mathcal{P}$, then $\mathbb{A}\in \mathcal{P}$ for all $\mathbb{A},\mathbb{B}\in 2^{\mathrm{\Xi }}$;

(P3)

If $(\mathbb{A}\cap \mathbb{B})\in \mathcal{P}$, then $\mathbb{A}\in \mathcal{P}$ or $\mathbb{B}\in \mathcal{P}$ for all $\mathbb{A},\mathbb{B}\in 2^{\mathrm{\Xi }}$.

Recall that the strategy for defining the filter structure $\mathcal{F}$ and the ideal structure $\mathcal{I}$ on a set $\mathrm{\Xi }$ looks like a fishing net, where some fish (subsets of $\mathrm{\Xi }$) did not get inside the net (are in the filter) and some other fish get inside the net (are in the ideal). If the smallest fish ($\varphi$) is filtered, then all bigger fish will be filtered and thus the net is useless. So, the proposed fish $\varphi$ could not be in the filter, $\varphi \notin \mathcal{F}$ (F1). If the largest fish ($\mathrm{\Xi }$) is in the ideal, then all smaller fish will be in the ideal and that is logically not good. So, the proposed fish $\mathrm{\Xi }$ could not be in the ideal, $\mathrm{\Xi }\notin \mathcal{I}$ (I1). For any fish that did not get inside the net, we assume that any larger fish also did not get inside (F2). The minimum ($\mathbb{A}\cap \mathbb{B}$) of two fish $\mathbb{A},\mathbb{B}$ in $\mathcal{F}$ (already filtered) will be filtered as well (F3). For any fish that get inside the net, we assume that any smaller fish should be inside the net (I2). The maximum ($\mathbb{A}\cup \mathbb{B}$) of two fish $\mathbb{A},\mathbb{B}$ in $\mathcal{I}$ (both get inside the net) will be in the ideal as well (I3).

This means that (F1) and (F2) (resp. (G1) and (G2)) yield to include the greater element in the filter (resp. grill), while (F3) (resp. (G3)) yields to include the smaller element in the filter (resp. grill). Also, (I1) and (I2) (resp. (P1) and (GP)) yield to include the smaller element in the ideal (resp. primal), while (I3) (resp. (P3)) yields to include the greater element in the ideal (resp. primal). Even in the definition of topology, (T2) yields to include the smaller element in the topology, while (T3) yields to include the greater element in the topology. All the other presented structures in this summary (generalized weak structure, supra topology, minimal structure, generalized topology, weak structure, infra topology, and anti-topology) are just generated definitions from the essential definition of topology, made only by making changes or restricting the axioms (T1)–(T3) of a general topology. That means that these generated definitions have no specific motivation to their concepts. However, all the definitions given in the summary are constructed without figuring out whether any specific property of the subsets satisfying the conditions of any of these definitions.

In our new definition of diving structures, we will construct the diving structures according to a specific nature that will allow the subsets to be included. This specific nature includes the largest subsets possible from all subsets of $P\left(\mathrm{\Xi }\right)$. Also, in our new definition of floating structures, we will construct the floating structures according to a specific nature that allows the subsets to be included. This specific nature is including the smallest subsets possible from all subsets of $P\left(\mathrm{\Xi }\right)$. So, our new structures have at least a trend or a clear property for the included subsets in both of these diving structures and floating structures.

Now, we introduce a new strategy for describing the subsets of $\mathrm{\Xi }$ that is different from all the previous notions in the summary.

Definition 1.

A non-empty collection $\mathcal{DV}\subseteq 2^{\mathrm{\Xi }}$ of subsets of a non-empty set Ξ is said to be a diving structure on Ξ if it satisfies the following conditions:

(DV1) 

$\mathrm{\Xi }\in \mathcal{DV}$

(DV2) 

$\varphi \notin \mathcal{DV}$,

(DV3) 

If $\mathbb{A}\in \mathcal{DV}$ and $\mathbb{B}\in \mathcal{DV}$, then $(\mathbb{A}\cup \mathbb{B})\in \mathcal{DV}$ for all $\mathbb{A},\mathbb{B}\in 2^{\mathrm{\Xi }}$,

(DV4) 

If $(\mathbb{A}\cap \mathbb{B})\in \mathcal{DV}$, then $\mathbb{A}\in \mathcal{DV}$ or $\mathbb{B}\in \mathcal{DV}$ for all $\mathbb{A},\mathbb{B}\in 2^{\mathrm{\Xi }}$.

(DVL4) is equivalent to

(DV4′) 

If $\mathbb{A}\notin \mathcal{DV}$ and $\mathbb{B}\notin \mathcal{DV}$, then $(\mathbb{A}\cap \mathbb{B})\notin \mathcal{DV}$ for all $\mathbb{A},\mathbb{B}\in 2^{\mathrm{\Xi }}$.

The pair $(\mathrm{\Xi },\mathcal{DV})$ is called a diving space.

Definition 2.

A non-empty collection $\mathcal{FL}\subseteq 2^{\mathrm{\Xi }}$ of subsets of a non-empty set Ξ is said to be a Floating structure on Ξ if it satisfies the following conditions:

(FL1) 

$\varphi \in \mathcal{FL}$,

(FL2) 

$\mathrm{\Xi }\notin \mathcal{FL}$,

(FL3) 

If $\mathbb{A}\in \mathcal{FL}$ and $\mathbb{B}\in \mathcal{FL}$, then $(\mathbb{A}\cap \mathbb{B})\in \mathcal{FL}$ for all $\mathbb{A},\mathbb{B}\in 2^{\mathrm{\Xi }}$,

(FL4) 

If $(\mathbb{A}\cup \mathbb{B})\in \mathcal{FL}$, then $\mathbb{A}\in \mathcal{FL}$ or $\mathbb{B}\in \mathcal{FL}$ for all $\mathbb{A},\mathbb{B}\in 2^{\mathrm{\Xi }}$.

(FL4) is equivalent to

(FL4′) 

If $\mathbb{A}\notin \mathcal{FL}$ and $\mathbb{B}\notin \mathcal{FL}$, then $(\mathbb{A}\cup \mathbb{B})\notin \mathcal{FL}$ for all $\mathbb{A},\mathbb{B}\in 2^{\mathrm{\Xi }}$.

The pair $(\mathrm{\Xi },\mathcal{FL})$ is called a floating space.

The conditions of a diving structure $\mathcal{DV}$ are equivalent to the conditions [$\mathrm{\Xi }\in \mathcal{DV}$, (G1), (I3) and (P3)]. The conditions of a floating structure $\mathcal{FL}$ are equivalent to the conditions [$\varphi \in \mathcal{FL}$, (P1), (F3) and (G3)].

Diving spaces must contain $\mathrm{\Xi }$ based on the heaviest weight that is diving to the bottom, and Floating spaces must contain $\varphi$ based on the lightest weight that is floating to shore. $\left\{\mathrm{\Xi }\right\}$ is the indiscrete diving structure and $\left\{\varphi \right\}$ is the indiscrete floating structure. $P\left(\mathrm{\Xi }\right)\setminus \left\{\varphi \right\}$ looks like the discrete diving structure and $P\left(\mathrm{\Xi }\right)\setminus \left\{\mathrm{\Xi }\right\}$ looks like the discrete floating structure. Both structures are not related to any of the previous structures in the summary. However, a floating structure could be a special type of the weak structure [13], and both the diving structures and floating structures could be special types of the generalized weak structures [8]. Moreover, both the diving spaces and floating spaces are not related to each other at all. Recall that $\left\{\varphi \right\}$ is an ideal but it fails to be a primal. Also, $\left\{\mathrm{\Xi }\right\}$ is a filter but it fails to be a grill. Notice that Not any non-empty structure coarser or finer than a diving structure is a diving structure as well, and not any structure coarser or finer than a floating structure is also floating structure.

All subsets of $\mathcal{DV}$ are called $\mathcal{DV}$-open sets in $\mathrm{\Xi }$, and all subsets of $\mathcal{FL}$ are called $\mathcal{FL}$-open sets in $\mathrm{\Xi }$.

${\mathcal{DV}}^c$ and ${\mathcal{FL}}^c$ denote the family of all $\mathcal{DV}$-closed sets and $\mathcal{FL}$-closed sets, respectively. Note that ${\mathcal{DV}}^c$ produces a floating structure and ${\mathcal{FL}}^c$ produces a diving structure. Hence, the complement of a $\mathcal{DV}$-open set is a $\mathcal{DV}$-closed set and must be an $\mathcal{FL}$-open set: that is, $\mathbb{A}$ is a $\mathcal{DV}$-open, ${\mathbb{A}}^c$ is $\mathcal{DV}$-closed, and ${\mathbb{A}}^c$ is an $\mathcal{FL}$-open set.

The complement of an $\mathcal{FL}$-open set is a $\mathcal{FL}$-closed set and must be a $\mathcal{DV}$-open set: that is, $\mathbb{A}$ is an $\mathcal{FL}$-open, ${\mathbb{A}}^c$ is $\mathcal{FL}$-closed, and ${\mathbb{A}}^c$ is a $\mathcal{DV}$-open set.

For any subset $\mathbb{A}\in 2^{\mathrm{\Xi }},\left|\mathrm{\Xi }\right|>2$ in a diving space $(\mathrm{\Xi },\mathcal{DV})$ or in a floating space $(\mathrm{\Xi },\mathcal{FL})$, it is impossible to find the cases $\mathcal{DV}=\{\mathrm{\Xi },\mathbb{A},{\mathbb{A}}^c\}$ and $\mathcal{FL}=\{\varphi ,\mathbb{A},{\mathbb{A}}^c\}$ because of conditions (DV4) and (FL4), respectively.

Definition 3.

The $\mathcal{DV}$-interior of a set $\mathbb{A}\subseteq \mathrm{\Xi }$, is defined as ${\mathbb{A}}_{\mathcal{DV}}^{\circ }=\cup \{\mathbb{B}\in \mathcal{DV}:\mathbb{B}\subseteq \mathbb{A}\}$, the $\mathcal{FL}$-interior of a set $\mathbb{A}\subseteq \mathrm{\Xi }$, is defined as ${\mathbb{A}}_{\mathcal{FL}}^{\circ }=\cup \{\mathbb{B}\in \mathcal{FL}:\mathbb{B}\subseteq \mathbb{A}\}$.

The $\mathcal{DV}$-closure of a set $\mathbb{A}\subseteq \mathrm{\Xi }$, is defined as $\overline{{\mathbb{A}}_{\mathcal{DV}}}=\cap \{\mathbb{B}\in {\mathcal{DV}}^c:\mathbb{B}\supseteq \mathbb{A}\}$, the $\mathcal{FL}$-closure of a set $\mathbb{A}\subseteq \mathrm{\Xi }$, is defined as $\overline{{\mathbb{A}}_{\mathcal{FL}}}=\cap \{\mathbb{B}\in {\mathcal{FL}}^c:\mathbb{B}\supseteq \mathbb{A}\}$.

$x\in \mathrm{\Xi }$ is called a $\mathcal{DV}$-interior point of a set $\mathbb{A}$$(x\in {\mathbb{A}}_{\mathcal{DV}}^{\circ })$ if $(\exists \mathbb{U}\in \mathcal{DV},x\in \mathbb{U})$$(\mathbb{U}\subseteq \mathbb{A})$, $x\in \mathrm{\Xi }$ is called an $\mathcal{FL}$-interior point of a set $\mathbb{A}$$(x\in {\mathbb{A}}_{\mathcal{FL}}^{\circ })$ if $(\exists \mathbb{U}\in \mathcal{FL},x\in \mathbb{U})$$(\mathbb{U}\subseteq \mathbb{A})$.

$x\in \mathrm{\Xi }$ is called a $\mathcal{DV}$-closure point of a set $\mathbb{A}$$(x\in \overline{{\mathbb{A}}_{\mathcal{DV}}})$ if $(\forall \mathbb{U}\in \mathcal{DV},x\in \mathbb{U})$$(\mathbb{U}\cap \mathbb{A}\ne \varphi )$, $x\in \mathrm{\Xi }$ is called an $\mathcal{FL}$-closure point of a set $\mathbb{A}$$(x\in \overline{{\mathbb{A}}_{\mathcal{FL}}})$ if $(\forall \mathbb{U}\in \mathcal{FL},x\in \mathbb{U})$$(\mathbb{U}\cap \mathbb{A}\ne \varphi )$.

$x\in \mathrm{\Xi }$ is called a $\mathcal{DV}$-limit point of a set $\mathbb{A}$ if for all $\mathbb{U}\in \mathcal{DV},x\in \mathbb{U}$, then $(\mathbb{U}\setminus \{x\left\}\right)\cap \mathbb{A}\ne \varphi$, $x\in \mathrm{\Xi }$ is called an $\mathcal{FL}$-limit point of a set $\mathbb{A}$ if for all $\mathbb{U}\in \mathcal{FL},x\in \mathbb{U}$, then $(\mathbb{U}\setminus \{x\left\}\right)\cap \mathbb{A}\ne \varphi$.

$\mathbb{A}$ is a $\mathcal{DV}$-neighborhood of $x\in \mathrm{\Xi }$ if $(\exists \mathbb{U}\in \mathcal{DV})$ so that $(x\in \mathbb{U}\subseteq \mathbb{A})$$(x\in {\mathbb{A}}_{\mathcal{DV}}^{\circ })$, $\mathbb{A}$ is an $\mathcal{FL}$-neighborhood of $x\in \mathrm{\Xi }$ if $(\exists \mathbb{U}\in \mathcal{FL})$ so that $(x\in \mathbb{U}\subseteq \mathbb{A})$$(x\in {\mathbb{A}}_{\mathcal{FL}}^{\circ })$.

Remark 1.

$\mathbb{A}\ne \varphi$ is a $\mathcal{DV}$-open if $\mathbb{A}={\mathbb{A}}_{\mathcal{DV}}^{\circ }$, while $\mathbb{A}$ is an $\mathcal{FL}$-open, implying that $\mathbb{A}={\mathbb{A}}_{\mathcal{FL}}^{\circ }$, but not the converse.

$\mathbb{A}\ne \mathrm{\Xi }$ is a $\mathcal{DV}$-closed if $\mathbb{A}=\overline{{\mathbb{A}}_{\mathcal{DV}}}$, while $\mathbb{A}$ is an $\mathcal{FL}$-closed, implying that $\mathbb{A}=\overline{{\mathbb{A}}_{\mathcal{FL}}}$ but not the converse.

${\left({\mathbb{A}}^c\right)}_{\mathcal{DV}}^{\circ }={\left(\overline{{\mathbb{A}}_{\mathcal{DV}}}\right)}^c$ and ${\left({\mathbb{A}}^c\right)}_{\mathcal{FL}}^{\circ }={\left(\overline{{\mathbb{A}}_{\mathcal{FL}}}\right)}^c$, $\overline{{\left({\mathbb{A}}^c\right)}_{\mathcal{DV}}}={\left({\mathbb{A}}_{\mathcal{DV}}^{\circ }\right)}^c$ and $\overline{{\left({\mathbb{A}}^c\right)}_{\mathcal{FL}}}={\left({\mathbb{A}}_{\mathcal{FL}}^{\circ }\right)}^c$.

Proposition 1.

(1) 

The interior operators ${\left(\right)}_{\mathcal{DV}}^{\circ }$ and ${\left(\right)}_{\mathcal{FL}}^{\circ }$ must satisfy ${\varphi }_{\mathcal{DV}}^{\circ }\ne \varphi$, while it is possible to find ${\mathrm{\Xi }}_{\mathcal{FL}}^{\circ }=\mathrm{\Xi }$.

(2) 

Usually, we have ${\mathrm{\Xi }}_{\mathcal{DV}}^{\circ }=\mathrm{\Xi }$, ${\varphi }_{\mathcal{FL}}^{\circ }=\varphi$.

(3) 

Clearly, $\mathbb{A}\subseteq \mathbb{B}$ implies that ${\mathbb{A}}_{\mathcal{DV}}^{\circ }\subseteq {\mathbb{B}}_{\mathcal{DV}}^{\circ }$ and ${\mathbb{A}}_{\mathcal{FL}}^{\circ }\subseteq {\mathbb{B}}_{\mathcal{FL}}^{\circ }$.

(4) 

${(\mathbb{A}\cap \mathbb{B})}_{\mathcal{DV}}^{\circ }={\mathbb{A}}_{\mathcal{DV}}^{\circ }\cap {\mathbb{B}}_{\mathcal{DV}}^{\circ }$ and ${(\mathbb{A}\cap \mathbb{B})}_{\mathcal{FL}}^{\circ }={\mathbb{A}}_{\mathcal{FL}}^{\circ }\cap {\mathbb{B}}_{\mathcal{FL}}^{\circ }$.

(5) 

Directly, we have for any $\mathbb{A}\subseteq \mathrm{\Xi }$, ${\left({\mathbb{A}}_{\mathcal{DV}}^{\circ }\right)}_{\mathcal{DV}}^{\circ }={\mathbb{A}}_{\mathcal{DV}}^{\circ }$, ${\left({\mathbb{A}}_{\mathcal{FL}}^{\circ }\right)}_{\mathcal{FL}}^{\circ }={\mathbb{A}}_{\mathcal{FL}}^{\circ }$.

Proposition 2.

(1) 

The closure operators $\overline{{\left(\right)}_{\mathcal{DV}}}$ and $\overline{{\left(\right)}_{\mathcal{FL}}}$ must satisfy $\overline{{\mathrm{\Xi }}_{\mathcal{DV}}}\ne \mathrm{\Xi }$, while it is possible to find $\overline{{\varphi }_{\mathcal{FL}}}=\varphi$.

(2) 

Usually, we have $\overline{{\varphi }_{\mathcal{DV}}}=\varphi$ and $\overline{{\mathrm{\Xi }}_{\mathcal{FL}}}=\mathrm{\Xi }$.

(3) 

Clearly, $\mathbb{A}\subseteq \mathbb{B}$ implies that $\overline{{\mathbb{A}}_{\mathcal{DV}}}\subseteq \overline{{\mathbb{B}}_{\mathcal{DV}}}$ and $\overline{{\mathbb{A}}_{\mathcal{FL}}}\subseteq \overline{{\mathbb{B}}_{\mathcal{FL}}}$.

(4) 

$\overline{{(\mathbb{A}\cup \mathbb{B})}_{\mathcal{DV}}}=\overline{{\mathbb{A}}_{\mathcal{DV}}}\cup \overline{{\mathbb{B}}_{\mathcal{DV}}}$ and $\overline{{(\mathbb{A}\cup \mathbb{B})}_{\mathcal{FL}}}=\overline{{\mathbb{A}}_{\mathcal{FL}}}\cup \overline{{\mathbb{B}}_{\mathcal{FL}}}$.

(5) 

For any $\mathbb{A}\subseteq \mathrm{\Xi }$, $\overline{{\left(\overline{{\mathbb{A}}_{\mathcal{DV}}}\right)}_{\mathcal{DV}}}=\overline{{\mathbb{A}}_{\mathcal{DV}}}$, $\overline{{\left(\overline{{\mathbb{A}}_{\mathcal{FL}}}\right)}_{\mathcal{FL}}}=\overline{{\mathbb{A}}_{\mathcal{FL}}}$.

(6) 

For any $\mathbb{A}\subseteq \mathrm{\Xi }$, ${\mathbb{A}}_{\mathcal{DV}}^{\circ }\subseteq \mathbb{A}\subseteq \overline{{\mathbb{A}}_{\mathcal{DV}}}$, ${\mathbb{A}}_{\mathcal{FL}}^{\circ }\subseteq \mathbb{A}\subseteq \overline{{\mathbb{A}}_{\mathcal{FL}}}$.

From the results given above for the operators ${\left(\right)}_{\mathcal{DV}}^{\circ }$, ${\left(\right)}_{\mathcal{FL}}^{\circ }$ and $\overline{{\left(\right)}_{\mathcal{DV}}}$, $\overline{{\left(\right)}_{\mathcal{FL}}}$, we can see that only ${\left(\right)}_{\mathcal{DV}}^{\circ }$ and $\overline{{\left(\right)}_{\mathcal{DV}}}$ are satisfying the common Kuratowski axioms of the interior and the closure operators in general topology; but, ${\left(\right)}_{\mathcal{FL}}^{\circ }$ and $\overline{{\left(\right)}_{\mathcal{FL}}}$ are not satisfying the common Kuratowski axioms of the interior and the closure operators in general topology. We explain this result in the following remark.

Remark 2.

Let $\mathrm{\Xi }=\{\varrho ,\varsigma ,\vartheta \}$. Then, $\mathcal{FL}=\{\varphi ,\{\varrho \},\{\varrho ,\varsigma \left\}\right\}$ is a floating structure on Ξ and $\mathcal{DV}=\{\mathrm{\Xi },\{\varrho \},\{\varrho ,\varsigma \left\}\right\}$ is a diving structure on Ξ, and, consequently, ${\mathcal{FL}}^c=\{\mathrm{\Xi },\{\varsigma ,\vartheta \},\left\{\vartheta \right\}\}$ and ${\mathcal{DV}}^c=\{\varphi ,\{\varsigma ,\vartheta \},\left\{\vartheta \right\}\}$. Now, ${\mathrm{\Xi }}_{\mathcal{DV}}^{\circ }=\mathrm{\Xi }$ but ${\mathrm{\Xi }}_{\mathcal{FL}}^{\circ }=\{\varrho ,\varsigma \}\ne \mathrm{\Xi }$. Moreover, $\overline{{\varphi }_{\mathcal{DV}}}=\varphi$ but $\overline{{\varphi }_{\mathcal{FL}}}=\left\{\vartheta \right\}\ne \varphi$. Hence, the first axiom $\overline{\varphi }=\varphi \mathrm{and} {\mathrm{\Xi }}^{\circ }=\mathrm{\Xi }$ of the Kuratowski axioms for both of interior and closure operators are not satisfied in case of the floating structures. In fact, the other three axioms of Kuratowski axioms are satisfied for both of the diving structures and the floating structures.

In a topological space $(\mathrm{\Xi },\tau )$ and a subset $\mathbb{A}\in 2^{\mathrm{\Xi }}$, a topological subspace was defined $(\mathbb{A},{\tau }_{\mathbb{A}})$, so that ${\tau }_{\mathbb{A}}=\{\mathbb{U}\cap \mathbb{A}:\mathbb{U}\in \tau \}$. Defining a diving subspace $(\mathbb{A},{\mathcal{DV}}_{\mathbb{A}})$ or a floating subspace $(\mathbb{A},{\mathcal{FL}}_{\mathbb{A}})$ in this way is not correct.

In the following example, we see that whenever a member of the space (subset in $P\left(\mathrm{\Xi }\right)$) gets larger, it dives down and the space succeeds to be a diving space, and whenever a member of the space (subset in $P\left(\mathrm{\Xi }\right)$) gets smaller, it floats up and the space succeeds to be a floating space. Note that conditions (DV3) and (DV4) take us down and conditions (FL3) and (FL4) take us up.

Example 1.

(a) 

Let $\mathrm{\Xi }=\left\{\varrho \right\}$ be a singleton set; then, $\left\{\varphi \right\}$ is a floating structure and $\left\{\mathrm{\Xi }\right\}$ is a diving structure. Both are the trivial cases not only for the singletons but for any non-empty set Ξ.

(b) 

If Ξ has two different elements, $\mathrm{\Xi }=\{\varrho ,\varsigma \}$, then $\{\varphi ,\{\varrho \left\}\right\}$, $\{\varphi ,\{\varsigma \left\}\right\}$ are floating structures on Ξ and $\{\mathrm{\Xi },\{\varrho \left\}\right\}$ and $\{\mathrm{\Xi },\{\varsigma \left\}\right\}$ are diving structures on Ξ. $\{\varphi ,\{\varrho \},\{\varsigma \left\}\right\}$ and $\{\mathrm{\Xi },\{\varrho \},\{\varsigma \left\}\right\}$ are the discrete floating structure and the discrete diving structure, respectively, on Ξ.

(c) 

If Ξ has three elements, $\mathrm{\Xi }=\{\varrho ,\varsigma ,\vartheta \}$, then $\{\varphi ,\{\varrho \left\}\right\}$, $\{\varphi ,\{\varsigma \left\}\right\}$, and $\{\varphi ,\{\vartheta \left\}\right\}$ are all floating structures but $\{\mathrm{\Xi },\{\varrho \left\}\right\}$, $\{\mathrm{\Xi },\{\varsigma \left\}\right\}$, and $\{\mathrm{\Xi },\{\vartheta \left\}\right\}$ are not all diving structures. Here, only condition (DV4) failed.

$\{\varphi ,\{\varrho \},\{\varsigma \left\}\right\}$, $\{\varphi ,\{\varrho \},\{\vartheta \left\}\right\}$, $\{\varphi ,\{\varsigma \},\{\vartheta \left\}\right\}$, and $\{\varphi ,\{\varrho \},\{\varsigma \},\{\vartheta \left\}\right\}$ are all floating structures but $\{\mathrm{\Xi },\{\varrho \},\{\varsigma \left\}\right\}$, $\{\mathrm{\Xi },\{\varrho \},\{\vartheta \left\}\right\}$, $\{\mathrm{\Xi },\{\varsigma \},\{\vartheta \left\}\right\}$, and $\{\mathrm{\Xi },\{\varrho \},\{\varsigma \},\{\vartheta \left\}\right\}$ are not all diving structures. Here, conditions (DV3) and (DV4) failed.

$\{\varphi ,\{\varrho \},\{\varrho ,\varsigma \left\}\right\},$$\{\varphi ,\{\vartheta \},\{\varrho ,\vartheta \left\}\right\}$, and $\{\varphi ,\{\varsigma \},\{\varsigma ,\vartheta \left\}\right\}$ are all floating structures and $\{\mathrm{\Xi },\{\varrho \},\{\varrho ,\varsigma \left\}\right\},$$\{\mathrm{\Xi },\{\vartheta \},\{\varrho ,\vartheta \left\}\right\}$, and $\{\mathrm{\Xi },\{\varsigma \},\{\varsigma ,\vartheta \left\}\right\}$ are all diving structures.

$\{\varphi ,\{\varrho \},\{\varsigma \},\{\varrho ,\varsigma \left\}\right\},$$\{\varphi ,\{\varrho \},\{\vartheta \},\{\varrho ,\vartheta \left\}\right\}$, and $\{\varphi ,\{\varsigma \},\{\vartheta \},\{\varsigma ,\vartheta \left\}\right\}$ are all floating structures, and $\{\mathrm{\Xi },\{\varrho \},\{\varsigma \},\{\varrho ,\varsigma \left\}\right\},$$\{\mathrm{\Xi },\{\varrho \},\{\vartheta \},\{\varrho ,\vartheta \left\}\right\}$, and $\{\mathrm{\Xi },\{\varsigma \},\{\vartheta \},\{\varsigma ,\vartheta \left\}\right\}$ are all diving structures.

$\{\mathrm{\Xi },\{\varrho ,\varsigma \left\}\right\}$, $\{\mathrm{\Xi },\{\varrho ,\vartheta \left\}\right\}$ and $\{\mathrm{\Xi },\{\varsigma ,\vartheta \left\}\right\}$ are all diving structures but $\{\varphi ,\{\varrho ,\varsigma \left\}\right\}$, $\{\varphi ,\{\varrho ,\vartheta \left\}\right\}$, and $\{\varphi ,\{\varsigma ,\vartheta \left\}\right\}$ are not all floating structures. Here, only condition (FL4) failed.

$\{\mathrm{\Xi },\{\varrho ,\varsigma \},\{\varrho ,\vartheta \left\}\right\}$, $\{\mathrm{\Xi },\{\varsigma ,\vartheta \},\{\varrho ,\vartheta \left\}\right\}$, and $\{\mathrm{\Xi },\{\varrho ,\varsigma \},\{\varsigma ,\vartheta \left\}\right\}$ are all diving structures but $\{\varphi ,\{\varrho ,\varsigma \},\{\varrho ,\vartheta \left\}\right\}$, $\{\varphi ,\{\varsigma ,\vartheta \},\{\varrho ,\vartheta \left\}\right\}$, and $\{\varphi ,\{\varrho ,\varsigma \},\{\varsigma ,\vartheta \left\}\right\}$ are not all floating structures. Here, conditions (FL3) and (FL4) failed.

$\{\varphi ,\{\varrho \},\{\varsigma ,\vartheta \left\}\right\}$, $\{\varphi ,\{\varsigma \},\{\varrho ,\vartheta \left\}\right\}$, and $\{\varphi ,\{\vartheta \},\{\varrho ,\varsigma \left\}\right\}$ are all not floating structures, and $\{\mathrm{\Xi },\{\varrho \},\{\varsigma ,\vartheta \left\}\right\}$, $\{\mathrm{\Xi },\{\varsigma \},\{\varrho ,\vartheta \left\}\right\}$ and $\{\mathrm{\Xi },\{\vartheta \},\{\varrho ,\varsigma \left\}\right\}$ are all not diving structures.

Now, for the floating structure $\mathcal{FL}=\{\varphi ,\{\varrho \},\{\varrho ,\vartheta \},\{\varrho ,\varsigma \left\}\right\}$, we obtain that ${\mathrm{\Xi }}_{\mathcal{FL}}^{\circ }=\mathrm{\Xi }$, while $\mathrm{\Xi }\notin \mathcal{FL}$. Also, from that ${\mathcal{FL}}^c=\{\mathrm{\Xi },\{\varsigma ,\vartheta \},\left\{\varsigma \right\},\left\{\vartheta \right\}\}$, we obtain that $\overline{{\varphi }_{\mathcal{FL}}}=\varphi$, while $\varphi \notin {\mathcal{FL}}^c$.

(d) 

If Ξ has four elements, $\mathrm{\Xi }=\{\varrho ,\varsigma ,\vartheta ,\varkappa \}$, then $\{\varphi ,\{\varrho \},\{\varrho ,\varsigma \left\}\right\}$ is a floating structure $\mathcal{FL}$, while the coarser structure $\{\varphi ,\{\varrho ,\varsigma \left\}\right\}$ is not a floating structure, and the finer structure $\{\varphi ,\{\varrho \},\{\varrho ,\varsigma \},\{\varsigma ,\vartheta \left\}\right\}$ is not a floating structure. Moreover, if we considered the subset $\mathbb{K}=\{\varrho ,\varsigma \}$ of the floating space $(\mathrm{\Xi },\mathcal{FL})$, we obtain that ${\mathcal{FL}}_{\mathbb{K}}=\{\mathbb{K},\left\{\varrho \right\},\varphi \}$, which is not a floating structure on $\mathbb{K}$.

Also, $\{\mathrm{\Xi },\{\varsigma ,\vartheta ,\varkappa \},\{\vartheta ,\varkappa \left\}\right\}$ is a diving structure $\mathcal{DV}$, while the coarser structure $\{\mathrm{\Xi },\{\vartheta ,\varkappa \left\}\right\}$ is not a diving structure, and the finer structure $\{\mathrm{\Xi },\{\varsigma ,\vartheta ,\varkappa \},\{\vartheta ,\varkappa \},\{\varrho \left\}\right\}$ is not a diving structure. Moreover, if we considered the subset $\mathbb{B}=\{\varrho ,\varsigma \}$ of the diving space $(\mathrm{\Xi },\mathcal{DV})$, we obtain that ${\mathcal{DV}}_{\mathbb{B}}=\{\mathbb{B},\left\{\varsigma \right\},\varphi \}$, which is not a diving structure on $\mathbb{B}$.

Now, define on a set $\mathrm{\Xi }=\{\varrho ,\varsigma ,\vartheta \}$ the floating structure $\mathcal{FL}=\{\varphi ,\{\varrho \},\{\varsigma \left\}\right\}$ and let $\mathbb{A}=\{\varrho ,\varsigma \}$; we obtain that ${\mathbb{A}}_{\mathcal{FL}}^{\circ }=\mathbb{A}$, while $\mathbb{A}\notin \mathcal{FL}$, that is, $\mathbb{A}$ is not $\mathcal{FL}$-open set. But, for any $\mathcal{FL}$-open set $\mathbb{A}$, we obtain that ${\mathbb{A}}_{\mathcal{FL}}^{\circ }=\mathbb{A}$. For the floating structure $\mathcal{FL}=\{\varphi ,\{\varrho \},\{\vartheta \left\}\right\}$ and $\mathbb{A}=\left\{\varsigma \right\}$, we obtain that $\overline{{\mathbb{A}}_{\mathcal{FL}}}=\mathbb{A}$, while $\mathbb{A}$ is not $\mathcal{FL}$-closed, that is, $\mathbb{A}\notin {\mathcal{FL}}^c$.

Example 2.

Let $\eta \in \mathrm{\Xi }$ and consider the excluded point topology $E_{\eta }=\left\{\mathrm{\Xi }\right\}\cup \{\mathbb{A}\subseteq \mathrm{\Xi }:\eta \notin \mathbb{A}\}$; then, $\{\mathbb{A}\subseteq \mathrm{\Xi }:\eta \notin \mathbb{A}\}$ is a floating structure on Ξ. Moreover, $\left\{\mathrm{\Xi }\right\}\cup \{\mathbb{A}\subseteq \mathrm{\Xi }:\eta \notin \mathbb{A},\mathbb{A}\ne \varphi \}$ is a diving structure on Ξ.

Consider the particular point topology $P_{\eta }=\left\{\varphi \right\}\cup \{\mathbb{A}\subseteq \mathrm{\Xi }:\eta \in \mathbb{A}\}$; then, $\{\mathbb{A}\subseteq \mathrm{\Xi }:\eta \in \mathbb{A}\}$ is a diving structure on Ξ. Moreover, $\left\{\varphi \right\}\cup \{\mathbb{A}\subseteq \mathrm{\Xi }:\eta \in \mathbb{A},\mathbb{A}\ne \mathrm{\Xi }\}$ is a floating structure on Ξ.

Example 3.

Let Ξ be a set of infinite elements and consider the cofinite topology $\mathcal{C}=\left\{\varphi \right\}\cup \{\mathbb{A}\subseteq \mathrm{\Xi }:{\mathbb{A}}^c is finite\}$; then, $\{\mathbb{A}\subseteq \mathrm{\Xi }:{\mathbb{A}}^c is finite\}$ is a diving structure on Ξ. Moreover, $\left\{\varphi \right\}\cup \{\mathbb{A}\subseteq \mathrm{\Xi }:{\mathbb{A}}^c is finite,\mathbb{A}\ne \mathrm{\Xi }\}$ is a floating structure on Ξ. This is the same with the cocountable topology ${\mathcal{C}}_n=\left\{\varphi \right\}\cup \{\mathbb{A}\subseteq \mathrm{\Xi }:{\mathbb{A}}^c is countable\}$; we have $\{\mathbb{A}\subseteq \mathrm{\Xi }:{\mathbb{A}}^c is countable\}$ is a diving structure on Ξ. Moreover, $\left\{\varphi \right\}\cup \{\mathbb{A}\subseteq \mathrm{\Xi }:{\mathbb{A}}^c is countable,\mathbb{A}\ne \mathrm{\Xi }\}$ is a floating structure on Ξ.

3. Separation Axioms in Diving Spaces and Floating Spaces

Separation axioms in a diving space $(\mathrm{\Xi },\mathcal{DV})$ or in a floating space $(\mathrm{\Xi },\mathcal{FL})$ could be defined as follows. Here, we introduce the lower axioms and the higher axioms, which could be defined easily by a common way.

Definition 4.

Let $(\mathrm{\Xi },\mathcal{DV})$ be a diving space (resp. a floating space $(\mathrm{\Xi },\mathcal{FL})$). Then,

(1) 

$(\mathrm{\Xi },\mathcal{DV})$ (resp. $(\mathrm{\Xi },\mathcal{FL})$) is called a $\mathcal{DV}$-$T_0$ (resp. $\mathcal{FL}$-$T_0$) if for every $\varrho \ne \varsigma \in \mathrm{\Xi }$, there exists $\mathbb{G}\subseteq \mathrm{\Xi }$ with $\varrho \in {\mathbb{G}}_{\mathcal{DV}}^{\circ }$ (resp. $\varrho \in {\mathbb{G}}_{\mathcal{FL}}^{\circ }$) so that $\varsigma \notin \mathbb{G}$ or there exists $\mathbb{H}\subseteq \mathrm{\Xi }$ with $\varsigma \in {\mathbb{H}}_{\mathcal{DV}}^{\circ }$ (resp. $\varsigma \in {\mathbb{H}}_{\mathcal{FL}}^{\circ }$) so that $\varrho \notin \mathbb{H}$.

(2) 

$(\mathrm{\Xi },\mathcal{DV})$ (resp. $(\mathrm{\Xi },\mathcal{FL})$) is called a $\mathcal{DV}$-$T_1$ (resp. $\mathcal{FL}$-$T_1$) if for every $\varrho \ne \varsigma \in \mathrm{\Xi }$, there exist $\mathbb{G},\mathbb{H}\subseteq \mathrm{\Xi }$ with $\varrho \in {\mathbb{G}}_{\mathcal{DV}}^{\circ }$, $\varsigma \in {\mathbb{H}}_{\mathcal{DV}}^{\circ }$ (resp. $\varrho \in {\mathbb{G}}_{\mathcal{FL}}^{\circ }$, $\varsigma \in {\mathbb{H}}_{\mathcal{FL}}^{\circ }$) so that $\varsigma \notin \mathbb{G}$ and $\varrho \notin \mathbb{H}$.

(3) 

$(\mathrm{\Xi },\mathcal{DV})$ (resp. $(\mathrm{\Xi },\mathcal{FL})$) is called a $\mathcal{DV}$-$T_2$ (resp. $\mathcal{FL}$-$T_2$) if for every $\varrho \ne \varsigma \in \mathrm{\Xi }$, there exist $\mathbb{G},\mathbb{H}\subseteq \mathrm{\Xi }$ with $\varrho \in {\mathbb{G}}_{\mathcal{DV}}^{\circ }$, $\varsigma \in {\mathbb{H}}_{\mathcal{DV}}^{\circ }$ (resp. $\varrho \in {\mathbb{G}}_{\mathcal{FL}}^{\circ }$, $\varsigma \in {\mathbb{H}}_{\mathcal{FL}}^{\circ }$) so that $\mathbb{G}\cap \mathbb{H}=\varphi$.

Remark 3.

Since there is no relation between $\mathcal{DV}$-interior and $\mathcal{FL}$-interior, thus we have

Definition 5.

A mapping $f:(\mathrm{\Xi },\mathcal{DV})\to ({\rm Y},{\mathcal{DV}}^{*})$ is said to be $\mathcal{DV}$-continuous if ${\left(f^{-1}\left(\mathbb{H}\right)\right)}_{\mathcal{DV}}^{\circ }\supseteq f^{-1}\left({\mathbb{H}}_{{\mathcal{DV}}^{*}}^{\circ }\right)\forall \mathbb{H}\subseteq {\rm Y},$ and a mapping $f:(\mathrm{\Xi },\mathcal{FL})\to ({\rm Y},{\mathcal{FL}}^{*})$ is said to be $\mathcal{FL}$-continuous if ${\left(f^{-1}\left(\mathbb{H}\right)\right)}_{\mathcal{FL}}^{\circ }\supseteq f^{-1}\left({\mathbb{H}}_{{\mathcal{FL}}^{*}}^{\circ }\right)\forall \mathbb{H}\subseteq {\rm Y}.$

It is equivalent to: A mapping $f:(\mathrm{\Xi },\mathcal{DV})\to ({\rm Y},{\mathcal{DV}}^{*})$ is $\mathcal{DV}$-continuous if $\overline{{\left(f^{-1}\left(\mathbb{H}\right)\right)}_{\mathcal{DV}}}\subseteq f^{-1}\left(\overline{{\mathbb{H}}_{{\mathcal{DV}}^{*}}}\right)\forall \mathbb{H}\subseteq {\rm Y},$ and a mapping $f:(\mathrm{\Xi },\mathcal{FL})\to ({\rm Y},{\mathcal{FL}}^{*})$ is $\mathcal{FL}$-continuous if $\overline{{\left(f^{-1}\left(\mathbb{H}\right)\right)}_{\mathcal{FL}}}\subseteq f^{-1}\left(\overline{{\mathbb{H}}_{{\mathcal{FL}}^{*}}}\right)\forall \mathbb{H}\subseteq {\rm Y}.$

Let us call $f:(\mathrm{\Xi },\mathcal{DV})\to ({\rm Y},{\mathcal{DV}}^{*})$$\mathcal{DV}$-open if ${\left(f\left(\mathbb{G}\right)\right)}_{{\mathcal{DV}}^{*}}^{\circ }\supseteq f\left({\mathbb{G}}_{\mathcal{DV}}^{\circ }\right)\forall \mathbb{G}\subseteq \mathrm{\Xi },$ $f:(\mathrm{\Xi },\mathcal{FL})\to ({\rm Y},{\mathcal{FL}}^{*})$$\mathcal{FL}$-open if ${\left(f\left(\mathbb{G}\right)\right)}_{{\mathcal{FL}}^{*}}^{\circ }\supseteq f\left({\mathbb{G}}_{\mathcal{FL}}^{\circ }\right)\forall \mathbb{G}\subseteq \mathrm{\Xi }.$

Also, let us call $f:(\mathrm{\Xi },\mathcal{DV})\to ({\rm Y},{\mathcal{DV}}^{*})$$\mathcal{DV}$-closed if $\overline{{\left(f\left(\mathbb{G}\right)\right)}_{{\mathcal{DV}}^{*}}}\subseteq f\left(\overline{{\mathbb{G}}_{\mathcal{DV}}}\right)\forall \mathbb{G}\subseteq \mathrm{\Xi },$ $f:(\mathrm{\Xi },\mathcal{FL})\to ({\rm Y},{\mathcal{FL}}^{*})$$\mathcal{FL}$-closed if $\overline{{\left(f\left(\mathbb{G}\right)\right)}_{{\mathcal{FL}}^{*}}}\subseteq f\left(\overline{{\mathbb{G}}_{\mathcal{FL}}}\right)\forall \mathbb{G}\subseteq \mathrm{\Xi }.$

Example 4.

(1) 

Note that, between topological spaces, the constant function is usually continuous but here, between diving spaces or floating spaces, it is not continuous. Let $f:(\mathrm{\Xi },\mathcal{DV})\to ({\rm Y},{\mathcal{DV}}^{*})$, $\mu \in {\rm Y}$ and $f\left(\varrho \right)=\mu \forall \varrho \in \mathrm{\Xi }$. Then, for any set $\mathbb{S}$ in ${\mathcal{DV}}^{*}$ with $\mu \notin \mathbb{S}$, we obtain $f^{-1}\left(\mathbb{S}\right)=\varphi \notin \mathcal{DV}$. Also, let $g:(\mathrm{\Xi },\mathcal{FL})\to ({\rm Y},{\mathcal{FL}}^{*})$, $\mu \in {\rm Y}$ and $g\left(\varrho \right)=\mu \forall \varrho \in \mathrm{\Xi }$. Then, for any set $\mathbb{H}$ in ${\mathcal{FL}}^{*}$ with $\mu \in \mathbb{H}$, we obtain $f^{-1}\left(\mathbb{H}\right)=\mathrm{\Xi }\notin \mathcal{FL}$. Moreover, f is $\mathcal{DV}$-open, $\mathcal{DV}$-closed whenever $\left\{\mu \right\}\in {\mathcal{DV}}^{*}$, and $\mathcal{FL}$-open and $\mathcal{FL}$-closed whenever $\left\{\mu \right\}\in {\mathcal{FL}}^{*}$.

(2) 

For $f:(\mathrm{\Xi },P\left(\mathrm{\Xi }\right)\setminus \left\{\varphi \right\})\to ({\rm Y},{\mathcal{DV}}^{*})$, where $P\left(\mathrm{\Xi }\right)\setminus \left\{\varphi \right\}$ is a discrete diving structure on Ξ, we obtain that f is $\mathcal{DV}$-continuous only if f was a surjective mapping, and this is different from the corresponding result in topological spaces (because for some $\mathbb{S}$ in ${\mathcal{DV}}^{*}$, we may obtain $f^{-1}\left(\mathbb{S}\right)=\varphi \notin \mathcal{DV}$). Similar restriction is required for the case between floating spaces. It is not necessary that f is $\mathcal{DV}$-open or $\mathcal{DV}$-closed or $\mathcal{FL}$-open or $\mathcal{FL}$-closed.

(3) 

For $f:(\mathrm{\Xi },\mathcal{DV})\to ({\rm Y},P({\rm Y})\setminus \{\varphi \left\}\right)$, where $P\left({\rm Y}\right)\setminus \left\{\varphi \right\}$ is a discrete diving structure on Υ, we obtain that f is $\mathcal{DV}$-open and $\mathcal{DV}$-closed (because for any $\mathbb{B}$ in $\mathcal{DV}$ (${\mathcal{DV}}^c$), we obtain $f\left(\mathbb{B}\right)\in {\mathcal{DV}}^{*}$ ($f\left(\mathbb{B}\right)\in {\left({\mathcal{DV}}^{*}\right)}^c$)). Similarly, it is true for the case between floating spaces. It is not necessary that f is $\mathcal{DV}$-continuous or $\mathcal{FL}$-continuous.

(4) 

For $f:(\mathrm{\Xi },\mathcal{DV})\to ({\rm Y},{\mathcal{DV}}^{*})$, where ${\mathcal{DV}}^{*}=\left\{{\rm Y}\right\}$ is the indiscrete diving structure on Υ, we obtain that f is $\mathcal{DV}$-continuous but not necessary to be $\mathcal{DV}$-open or $\mathcal{DV}$-closed (because may be for $\mathbb{B}$ in $\mathcal{DV}$ (${\mathcal{DV}}^c$), we obtain $f\left(\mathbb{B}\right)\notin {\mathcal{DV}}^{*}$ ($f\left(\mathbb{B}\right)\notin {\left({\mathcal{DV}}^{*}\right)}^c$)). Similarly, it is true for the case between floating spaces whenever we take ${\mathcal{FL}}^{*}=\left\{\varphi \right\}$.

(5) 

However, the identity mapping between diving spaces (resp. floating spaces) is usually $\mathcal{DV}$-continuous or $\mathcal{DV}$-open or $\mathcal{DV}$-closed (resp. $\mathcal{FL}$-continuous or $\mathcal{FL}$-open $\mathcal{FL}$-closed). The identity mapping here adopts an usual behavior in topological spaces.

In the following theorems, we show that, between diving spaces, the image and the inverse image of a $\mathcal{DV}$-$T_2$ space will be a $\mathcal{DV}$-$T_2$ space. The same propositions are clear and similar in the case between floating spaces.

Theorem 1.

Let $(\mathrm{\Xi },\mathcal{DV}),({\rm Y},{\mathcal{DV}}^{*})$ be two diving spaces and $f:(\mathrm{\Xi },\mathcal{DV})\to ({\rm Y},{\mathcal{DV}}^{*})$ an injective $\mathcal{DV}$-continuous mapping. Then, Ξ is a $\mathcal{DV}$-$T_i$ space if Υ is a ${\mathcal{DV}}^{*}$-$T_i$ space, $i=0,1,2$.

Proof. 

Since $\varrho \ne \varsigma$ in $\mathrm{\Xi }$ implies that $f\left(\varrho \right)\ne f\left(\varsigma \right)$ in ${\rm Y}$, and from ${\rm Y}$ is a ${\mathcal{DV}}^{*}$-$T_2$ space, then there exist $\mathbb{G},\mathbb{K}\in 2^{{\rm Y}}$ with $f\left(\varrho \right)\in {\mathbb{G}}_{{\mathcal{DV}}^{*}}^{\circ }$, $f\left(\varsigma \right)\in {\mathbb{K}}_{{\mathcal{DV}}^{*}}^{\circ }$ so that $\mathbb{G}\cap \mathbb{K}=\varphi$, that is, $\varrho \in f^{-1}\left({\mathbb{G}}_{{\mathcal{DV}}^{*}}^{\circ }\right)$, $\varsigma \in f^{-1}\left({\mathbb{K}}_{{\mathcal{DV}}^{*}}^{\circ }\right)$, and from that f is $\mathcal{DV}$-continuous, then $\varrho \in {\left(f^{-1}\left(\mathbb{G}\right)\right)}_{\mathcal{DV}}^{\circ }$ and $\varsigma \in {\left(f^{-1}\left(\mathbb{K}\right)\right)}_{\mathcal{DV}}^{\circ }$. That is, there exist $\mathbb{A}=f^{-1}\left(\mathbb{G}\right),\mathbb{B}=f^{-1}\left(\mathbb{K}\right)\in 2^{\mathrm{\Xi }}$ with $\varrho \in {\mathbb{A}}_{\mathcal{DV}}^{\circ }$ and $\varsigma \in {\mathbb{B}}_{\mathcal{DV}}^{\circ }$ so that $\mathbb{A}\cap \mathbb{B}=f^{-1}\left(\mathbb{G}\right)\cap f^{-1}\left(\mathbb{K}\right)=f^{-1}(\mathbb{G}\cap \mathbb{K})=\varphi$. Hence, $\mathrm{\Xi }$ is a $\mathcal{DV}$-$T_2$ space. Other cases are similar. □

Theorem 2.

Let $(\mathrm{\Xi },\mathcal{DV}),(\mathrm{{\rm Y}},{\mathcal{DV}}^{*})$ be two diving spaces and $f:(\mathrm{\Xi },\mathcal{DV})\to ({\rm Y},{\mathcal{DV}}^{*})$ a surjective $\mathcal{DV}$-open mapping. Then, Υ is a ${\mathcal{DV}}^{*}$-$T_i$ space if Ξ is a $\mathcal{DV}$-$T_i$ space, $i=0,1,2$.

Proof. 

Since $\mu \ne \nu$ in $\mathrm{{\rm Y}}$ implies that there are $\varrho \in f^{-1}\left(\mu \right)$ and $\varsigma \in f^{-1}\left(\nu \right)$, $f\left(\varrho \right)=\mu ,f\left(\varsigma \right)=\nu$ with $\varrho \ne \varsigma$ in $\mathrm{\Xi }$, and from $\mathrm{\Xi }$ is a $\mathcal{DV}$-$T_2$, then there exist $\mathbb{G},\mathbb{K}\in 2^{\mathrm{\Xi }}$ with $\varrho \in {\mathbb{G}}_{\mathcal{DV}}^{\circ }$, $\varsigma \in {\mathbb{K}}_{\mathcal{DV}}^{\circ }$ so that $\mathbb{G}\cap \mathbb{K}=\varphi$, that is, $\mu =f\left(\varrho \right)\in f\left({\mathbb{G}}_{\mathcal{DV}}^{\circ }\right)$, $\nu =f\left(\varsigma \right)\in f\left({\mathbb{K}}_{\mathcal{DV}}^{\circ }\right)$, and from that f is $\mathcal{DV}$-open, then $\mu \in {\left(f\left(\mathbb{G}\right)\right)}_{{\mathcal{DV}}^{*}}^{\circ }$ and $\nu \in {\left(f\left(\mathbb{K}\right)\right)}_{{\mathcal{DV}}^{*}}^{\circ }$. That is, there exist $\mathbb{A}=f\left(\mathbb{G}\right),\mathbb{B}=f\left(\mathbb{K}\right)\in 2^{{\rm Y}}$ with $\mu \in {\mathbb{A}}_{{\mathcal{DV}}^{*}}^{\circ }$ and $\nu \in {\mathbb{B}}_{{\mathcal{DV}}^{*}}^{\circ }$ so that $\mathbb{A}\cap \mathbb{B}=f\left(\mathbb{G}\right)\cap f\left(\mathbb{K}\right)\subseteq f(\mathbb{G}\cap \mathbb{K})=\varphi$. Hence, ${\rm Y}$ is a ${\mathcal{DV}}^{*}$-$T_2$ space. Other cases are similar. □

Example 5.

Consider a set $\mathrm{\Xi }=\{\varrho ,\varsigma \}$. Then, the floating structures $\{\varphi ,\{\varrho \left\}\right\}$ and $\{\varphi ,\{\varsigma \left\}\right\}$ are examples for an $\mathcal{FL}$-$T_0$ but not $\mathcal{FL}$-$T_1$ and not $\mathcal{FL}$-$T_2$. Also, the diving structures $\{\mathrm{\Xi },\{\varrho \left\}\right\}$ and $\{\mathrm{\Xi },\{\varsigma \left\}\right\}$ are examples for $\mathcal{DV}$-$T_0$ but not $\mathcal{DV}$-$T_1$ and not $\mathcal{DV}$-$T_2$. The discrete floating structure $\{\varphi ,\{\varrho \},\{\varsigma \left\}\right\}$ and the discrete diving structure $\{\mathrm{\Xi },\{\varrho \},\{\varsigma \left\}\right\}$ are samples of $\mathcal{DV}$-$T_2$ structures and $\mathcal{FL}$-$T_2$ structures, respectively.

Example 6.

From Example 3, in which Ξ is an infinite set, we obtain an example for a $\mathcal{DV}$-$T_1$ space but not a $\mathcal{DV}$-$T_2$ space. Clearly, there are $\mathrm{\Xi }\setminus \left\{x\right\}$, $\mathrm{\Xi }\setminus \left\{y\right\}$ as two neighborhoods in $\mathcal{DV}$ satisfying the axiom $\mathcal{DV}$-$T_1$. But, if we supposed that it is a $\mathcal{DV}$-$T_2$, then there are $U,V\in \mathcal{DV}$, such that $x\in U,y\in V$ and $U\cap V=\varphi$. Since $U^c\cup V^c={(U\cap V)}^c={\varphi }^c=\mathrm{\Xi }$, Ξ is an infinite set while $U^c$ is finite and $V^c$ is finite, then $U^c\cup V^c$ is finite and we find a contradiction. Hence, $(\mathrm{\Xi },\mathcal{DV})$ could not be $\mathcal{DV}$-$T_2$ space.

This is similar when showing the case for the floating structure.

4. Connected Diving Spaces and Floating Spaces

Definition 6.

If $(\mathrm{\Xi },\mathcal{DV})$ is a diving space and $(\mathrm{\Xi },\mathcal{FL})$ is a floating space, then the following criteria must be fulfilled:

(1) 

Two sets $\mathbb{B},\mathbb{C}\in 2^{\mathrm{\Xi }}$ are called $\mathcal{DV}$-separated (resp. $\mathcal{FL}$-separated) if

$$\overline{{\mathbb{B}}_{\mathcal{DV}}}\cap \mathbb{C}=\mathbb{B}\cap \overline{{\mathbb{C}}_{\mathcal{DV}}}=\varphi (resp.\overline{{\mathbb{B}}_{\mathcal{FL}}}\cap \mathbb{C}=\mathbb{B}\cap \overline{{\mathbb{C}}_{\mathcal{FL}}}=\varphi ).$$

(2) 

A set $\mathbb{G}\in 2^{\mathrm{\Xi }}$ is called $\mathcal{DV}$-disconnected (resp. $\mathcal{FL}$-disconnected) set if there exist $\mathcal{DV}$-separated (resp. $\mathcal{FL}$-separated) sets $\mathbb{B},\mathbb{C}\in 2^{\mathrm{\Xi }}$, so that $\mathbb{B}\cup \mathbb{C}=\mathbb{G}$.

A set $\mathbb{G}$ is called $\mathcal{DV}$-connected (resp. $\mathcal{FL}$-connected) if it is not $\mathcal{DV}$-disconnected (resp. not $\mathcal{FL}$-disconnected).

(3) 

The space $(\mathrm{\Xi },\mathcal{DV})$ is called a $\mathcal{DV}$-disconnected space if there exist $\mathcal{DV}$-separated sets $\mathbb{B},\mathbb{C}\in 2^{\mathrm{\Xi }}$, so that $\mathbb{B}\cup \mathbb{C}=\mathrm{\Xi }$.

A space $(\mathrm{\Xi },\mathcal{FL})$ is called $\mathcal{FL}$-disconnected space if there exist $\mathcal{FL}$-separated sets $\mathbb{B},\mathbb{C}\in 2^{\mathrm{\Xi }}$ so that $\mathbb{B}\cup \mathbb{C}=\mathrm{\Xi }$.

(4) 

The space $(\mathrm{\Xi },\mathcal{DV})$ is called $\mathcal{DV}$-connected space if there are no $\mathcal{DV}$-separated sets $\mathbb{B},\mathbb{C}\in 2^{\mathrm{\Xi }}$, so that $\mathbb{B}\cup \mathbb{C}=\mathrm{\Xi }$ except $\mathbb{B}=\varphi$ or $\mathbb{C}=\varphi$.

A space $(\mathrm{\Xi },\mathcal{FL})$ is called $\mathcal{FL}$-connected space if there are no $\mathcal{FL}$-separated sets $\mathbb{B},\mathbb{C}\in 2^{\mathrm{\Xi }}$ so that $\mathbb{B}\cup \mathbb{C}=\mathrm{\Xi }$ except $\mathbb{B}=\varphi$ or $\mathbb{C}=\varphi$.

Remark 4.

Since there is no relation between $\overline{{\mathbb{B}}_{\mathcal{DV}}}$ and $\overline{{\mathbb{B}}_{\mathcal{FL}}}$ for any subset $\mathbb{B}\in 2^{\mathrm{\Xi }}$, then there is no relation between connectedness in diving spaces and connectedness in floating spaces.

In a disconnected diving space $(\mathrm{\Xi },\mathcal{DV})$, any set $\mathbb{A}\in 2^{\mathrm{\Xi }}$ will be a $\mathcal{DV}$-disconnected set. Also, in a disconnected floating space $(\mathrm{\Xi },\mathcal{FL})$, any subset $\mathbb{A}\in 2^{\mathrm{\Xi }}$ will be an $\mathcal{FL}$-disconnected set. Conversely, any super set $\mathbb{A}\in 2^{\mathrm{\Xi }}$ of a $\mathcal{DV}$-connected space $(\mathrm{\Xi },\mathcal{DV})$ ($\mathcal{FL}$-connected space $(\mathrm{\Xi },\mathcal{FL})$) is a $\mathcal{DV}$-connected (an $\mathcal{FL}$-connected) set as well.

Example 7.

If $\mathrm{\Xi }=\{\varrho ,\varsigma \}$, then the floating structures $\{\varphi ,\{\varrho \left\}\right\}$ and $\{\varphi ,\{\varsigma \left\}\right\}$ are examples of an $\mathcal{FL}$-connected structure while $\{\varphi ,\{\varrho \},\{\varsigma \left\}\right\}$ is an $\mathcal{FL}$-disconnected structure. The diving structures $\{\mathrm{\Xi },\{\varrho \left\}\right\}$ and $\{\mathrm{\Xi },\{\varsigma \left\}\right\}$ are examples for a $\mathcal{DV}$-connected structure while $\{\mathrm{\Xi },\{\varrho \},\{\varsigma \left\}\right\}$ is a $\mathcal{DV}$-disconnected structure. That is, the finer structure of a $\mathcal{DV}$-disconnected (an $\mathcal{FL}$-disconnected) structure will be a $\mathcal{DV}$-disconnected (an $\mathcal{FL}$-disconnected) as well. But, the coarser structure of a $\mathcal{DV}$-disconnected (an $\mathcal{FL}$-disconnected) structure maybe a $\mathcal{DV}$-connected (an $\mathcal{FL}$-connected) structure.

Example 8.

As given in Example 3, we obtain that $\mathcal{DV}=\{\mathbb{A}\subseteq \mathrm{\Xi }:{\mathbb{A}}^c is finite\}$ is a diving structure on Ξ, and then ${\mathcal{DV}}^c=\{\mathbb{A}\subseteq \mathrm{\Xi }:{\mathbb{A}}^c is infinite\}$. Moreover, $\mathcal{FL}=\left\{\varphi \right\}\cup \{\mathbb{A}\subseteq \mathrm{\Xi }:{\mathbb{A}}^c is finite,\mathbb{A}\ne \mathrm{\Xi }\}$ is a floating structure on Ξ, and then ${\mathcal{FL}}^c=\left\{\mathrm{\Xi }\right\}\cup \{\mathbb{A}\subseteq \mathrm{\Xi }:{\mathbb{A}}^c is infinite,\mathbb{A}\ne \varphi \}$. Now, let $\mathrm{\Xi }=\mathbb{R}$, $H=\mathbb{Q},K={\mathbb{Q}}^c$, then $\overline{{\mathbb{H}}_{\mathcal{DV}}}=H=\mathbb{Q}$, $\overline{{\mathbb{K}}_{\mathcal{DV}}}=K={\mathbb{Q}}^c$, and also $\overline{{\mathbb{H}}_{\mathcal{FL}}}=H=\mathbb{Q}$, $\overline{{\mathbb{K}}_{\mathcal{FL}}}=K={\mathbb{Q}}^c$. Thus, H and K are two separated sets in $(\mathbb{R},\mathcal{DV})$ and $(\mathbb{R},\mathcal{FL})$, such that $H\cup K=\mathrm{\Xi }=\mathbb{R}$, and hence, both diving and floating spaces are disconnected spaces.

5. Roughness in Diving Spaces and Floating Spaces

Let $\mathbb{R}$ be an equivalence relation on $\mathrm{\Xi }$; then, there is a partitioning of $\mathrm{\Xi }$ by the equivalence classes $\left[\varrho \right]\forall \varrho \in \mathrm{\Xi }$. These equivalence classes have the main role in defining the lower approximation set ${\mathbb{A}}_{\mathbb{R}}$ and the upper approximation set ${\mathbb{A}}^{\mathbb{R}}$ of some subset $\mathbb{A}\in 2^{\mathrm{\Xi }}$ in both of diving spaces $(\mathrm{\Xi },\mathcal{DV})$ and floating spaces $(\mathrm{\Xi },\mathcal{FL})$.

For a subset $\mathbb{A}\in 2^{\mathrm{\Xi }}$ in the diving space $(\mathrm{\Xi },\mathcal{DV})$, define the approximation subsets ${\left({\mathbb{A}}_{*}\right)}^{\mathcal{DV}},{\left({\mathbb{A}}^{*}\right)}_{\mathcal{DV}}\in 2^{\mathrm{\Xi }}$ by

$${\left({\mathbb{A}}_{*}\right)}^{\mathcal{DV}}=\{\varrho \in \mathrm{\Xi }:\left[\varrho \right]\cap {\mathbb{A}}^c\in {\mathcal{DV}}^c\},{\left({\mathbb{A}}^{*}\right)}_{\mathcal{DV}}=\{\varrho \in \mathrm{\Xi }:\left[\varrho \right]\cap \mathbb{A}\notin {\mathcal{DV}}^c\}.$$

(1)

For a subset $\mathbb{A}\in 2^{\mathrm{\Xi }}$ in the floating space $(\mathrm{\Xi },\mathcal{FL})$, define the approximation subsets ${\left({\mathbb{A}}_{*}\right)}^{\mathcal{FL}},{\left({\mathbb{A}}^{*}\right)}_{\mathcal{FL}}\in 2^{\mathrm{\Xi }}$ by

$${\left({\mathbb{A}}_{*}\right)}^{\mathcal{FL}}=\{\varrho \in \mathrm{\Xi }:\left[\varrho \right]\cap {\mathbb{A}}^c\in \mathcal{FL}\},{\left({\mathbb{A}}^{*}\right)}_{\mathcal{FL}}=\{\varrho \in \mathrm{\Xi }:\left[\varrho \right]\cap \mathbb{A}\notin \mathcal{FL}\}.$$

(2)

For any subset $\mathbb{A}\in 2^{\mathrm{\Xi }}$ of $(\mathrm{\Xi },\mathcal{DV})$, the diving lower approximation${\left({\mathbb{A}}_{\mathbb{R}}\right)}^{\mathcal{DV}}$ and the diving upper approximation${\left({\mathbb{A}}^{\mathbb{R}}\right)}_{\mathcal{DV}}$ are defined by:

$${\left({\mathbb{A}}_{\mathbb{R}}\right)}^{\mathcal{DV}}=\mathbb{A}\cap {\left({\mathbb{A}}_{*}\right)}^{\mathcal{DV}},{\left({\mathbb{A}}^{\mathbb{R}}\right)}_{\mathcal{DV}}=\mathbb{A}\cup {\left({\mathbb{A}}^{*}\right)}_{\mathcal{DV}}.$$

(3)

For any subset $\mathbb{A}\in 2^{\mathrm{\Xi }}$ of $(\mathrm{\Xi },\mathcal{FL})$, the floating lower approximation${\left({\mathbb{A}}_{\mathbb{R}}\right)}^{\mathcal{FL}}$ and the floating upper approximation${\left({\mathbb{A}}^{\mathbb{R}}\right)}_{\mathcal{FL}}$ are defined by:

$${\left({\mathbb{A}}_{\mathbb{R}}\right)}^{\mathcal{FL}}=\mathbb{A}\cap {\left({\mathbb{A}}_{*}\right)}^{\mathcal{FL}},{\left({\mathbb{A}}^{\mathbb{R}}\right)}_{\mathcal{FL}}=\mathbb{A}\cup {\left({\mathbb{A}}^{*}\right)}_{\mathcal{FL}}.$$

(4)

The boundary region sets ${\left({\mathbb{A}}^{\mathbb{B}}\right)}_{\mathcal{DV}}$ and ${\left({\mathbb{A}}^{\mathbb{B}}\right)}_{\mathcal{FL}}$ are defined by the set differences ${\left({\mathbb{A}}^{\mathbb{R}}\right)}_{\mathcal{DV}}\setminus {\left({\mathbb{A}}_{\mathbb{R}}\right)}^{\mathcal{DV}}={\left({\mathbb{A}}^{\mathbb{B}}\right)}_{\mathcal{DV}}$ and ${\left({\mathbb{A}}^{\mathbb{R}}\right)}_{\mathcal{FL}}\setminus {\left({\mathbb{A}}_{\mathbb{R}}\right)}^{\mathcal{FL}}={\left({\mathbb{A}}^{\mathbb{B}}\right)}_{\mathcal{FL}}$, respectively.

Moreover, the accuracy value ${\mathbb{A}}^{\alpha }$ of a rough set $\mathbb{A}$ in a diving space $(\mathrm{\Xi },\mathcal{DV})$ (resp. a floating space $(\mathrm{\Xi },\mathcal{FL})$) is given by the fraction

$${\left({\mathbb{A}}^{\alpha }\right)}_{\mathcal{DV}}={\displaystyle \frac{no. of elements of {\left({\mathbb{A}}_{\mathbb{R}}\right)}^{\mathcal{DV}}}{no. of elements of {\left({\mathbb{A}}^{\mathbb{R}}\right)}_{\mathcal{DV}}}}(\mathrm{resp}.{\left({\mathbb{A}}^{\alpha }\right)}_{\mathcal{FL}}={\displaystyle \frac{no. of elements of {\left({\mathbb{A}}_{\mathbb{R}}\right)}^{\mathcal{FL}}}{no. of elements of {\left({\mathbb{A}}^{\mathbb{R}}\right)}_{\mathcal{FL}}}}).$$

The triple $(\mathrm{\Xi },\mathbb{R},\mathcal{DV})$ is called a diving approximation space and the triple $(\mathrm{\Xi },\mathbb{R},\mathcal{FL})$ is called a floating approximation space. In any space $(\mathrm{\Xi },\mathbb{R},\mathcal{DV})$ or $(\mathrm{\Xi },\mathbb{R},\mathcal{FL})$, whenever ${\mathbb{A}}^{\mathbb{R}}⊈{\mathbb{A}}_{\mathbb{R}},$ then ${\mathbb{A}}^{\mathbb{B}}$ will be not empty and the set $\mathbb{A}$ is called rough set. As a special case, if ${\mathbb{A}}^{\mathbb{R}}=\mathrm{\Xi },{\mathbb{A}}_{\mathbb{R}}=\varphi$; then, ${\mathbb{A}}^{\mathbb{B}}=\mathrm{\Xi }$ and $\mathbb{A}$ is called totally rough set. If ${\mathbb{A}}^{\mathbb{R}}\subseteq {\mathbb{A}}_{\mathbb{R}}$, then ${\mathbb{A}}^{\mathbb{B}}=\varphi$ and $\mathbb{A}$ is called exact set.

Remark 5.

In case of the indiscrete diving structure $\left\{\mathrm{\Xi }\right\}$, we obtain that ${\left({\mathbb{A}}_{*}\right)}^{\mathcal{DV}}=\mathrm{\Xi }$ and ${\left({\mathbb{A}}^{*}\right)}_{\mathcal{DV}}=\varphi$ in general, and so ${\left({\mathrm{\Xi }}_{*}\right)}^{\mathcal{DV}}={\left({\varphi }_{*}\right)}^{\mathcal{DV}}=\mathrm{\Xi }$, while ${\left({\varphi }^{*}\right)}_{\mathcal{DV}}=\varphi$ whenever $\left[\varrho \right]\ne \mathrm{\Xi }$ for all $\varrho \in \mathrm{\Xi }$ and ${\left({\mathrm{\Xi }}^{*}\right)}_{\mathcal{DV}}=\mathrm{\Xi }$ whenever $\left[\varrho \right]=\mathrm{\Xi }$ for some $\varrho \in \mathrm{\Xi }$. Moreover, in case of the indiscrete floating structure $\left\{\varphi \right\}$, ${\left({\mathbb{A}}_{*}\right)}^{\mathcal{FL}}\subseteq \mathbb{A}\subseteq {\left({\mathbb{A}}^{*}\right)}_{\mathcal{FL}}$ in general except ${\left({\mathrm{\Xi }}_{*}\right)}^{\mathcal{FL}}={\left({\mathrm{\Xi }}^{*}\right)}_{\mathcal{FL}}=\mathrm{\Xi }$ and ${\left({\varphi }_{*}\right)}^{\mathcal{FL}}={\left({\varphi }^{*}\right)}_{\mathcal{FL}}=\varphi$.

Lemma 1.

Let $\mathbb{R}$ be an equivalence relation on Ξ, and $\mathbb{A},\mathbb{B}\in P\left(\mathrm{\Xi }\right)$. Then, the following properties hold:

(1)

${\left({\mathbb{A}}^{*}\right)}_{\mathcal{DV}}={\left({\left({\mathbb{A}}^c\right)}_{*}^{\mathcal{DV}}\right)}^c$ and ${\left({\mathbb{A}}^{*}\right)}_{\mathcal{FL}}={\left({\left({\mathbb{A}}^c\right)}_{*}^{\mathcal{FL}}\right)}^c$,

(2)

$\mathbb{A}\subseteq \mathbb{B}$ implies that ${\left({\mathbb{A}}_{*}\right)}^{\mathcal{DV}}\subseteq {\left({\mathbb{B}}_{*}\right)}^{\mathcal{DV}}$, ${\left({\mathbb{A}}_{*}\right)}^{\mathcal{FL}}\subseteq {\left({\mathbb{B}}_{*}\right)}^{\mathcal{FL}}$,

(3)

$\mathbb{A}\subseteq \mathbb{B}$ implies that ${\left({\mathbb{A}}^{*}\right)}_{\mathcal{DV}}\subseteq {\left({\mathbb{B}}^{*}\right)}_{\mathcal{DV}}$, ${\left({\mathbb{A}}^{*}\right)}_{\mathcal{FL}}\subseteq {\left({\mathbb{B}}^{*}\right)}_{\mathcal{FL}}$,

(4)

${(\mathbb{A}\cap \mathbb{B})}_{\mathcal{DV}}^{*}\subseteq {\left({\mathbb{A}}^{*}\right)}_{\mathcal{DV}}\cap {\left({\mathbb{B}}^{*}\right)}_{\mathcal{DV}}$,

(5)

${(\mathbb{A}\cap \mathbb{B})}_{\mathcal{FL}}^{*}\subseteq {\left({\mathbb{A}}^{*}\right)}_{\mathcal{FL}}\cap {\left({\mathbb{B}}^{*}\right)}_{\mathcal{FL}}$,

(6)

${(\mathbb{A}\cap \mathbb{B})}_{*}^{\mathcal{DV}}\subseteq {\left({\mathbb{A}}_{*}\right)}^{\mathcal{DV}}\cap {\left({\mathbb{B}}_{*}\right)}^{\mathcal{DV}}$,

(7)

${(\mathbb{A}\cap \mathbb{B})}_{*}^{\mathcal{FL}}\subseteq {\left({\mathbb{A}}_{*}\right)}^{\mathcal{FL}}\cap {\left({\mathbb{B}}_{*}\right)}^{\mathcal{FL}}$,

(8)

${(\mathbb{A}\cup \mathbb{B})}_{\mathcal{DV}}^{*}\supseteq {\left({\mathbb{A}}^{*}\right)}_{\mathcal{DV}}\cup {\left({\mathbb{B}}^{*}\right)}_{\mathcal{DV}}$,

(9)

${(\mathbb{A}\cup \mathbb{B})}_{\mathcal{FL}}^{*}\supseteq {\left({\mathbb{A}}^{*}\right)}_{\mathcal{FL}}\cup {\left({\mathbb{B}}^{*}\right)}_{\mathcal{FL}}$,

(10)

${(\mathbb{A}\cup \mathbb{B})}_{*}^{\mathcal{DV}}\supseteq {\left({\mathbb{A}}_{*}\right)}^{\mathcal{DV}}\cup {\left({\mathbb{B}}_{*}\right)}^{\mathcal{DV}}$,

(11)

${(\mathbb{A}\cup \mathbb{B})}_{*}^{\mathcal{FL}}\supseteq {\left({\mathbb{A}}_{*}\right)}^{\mathcal{FL}}\cup {\left({\mathbb{B}}_{*}\right)}^{\mathcal{FL}}$.

Proof. 

From Equations (1) and (2), it is clear that ${\left({\mathbb{A}}_{\mathcal{DV}}^{*}\right)}^c={\left({\mathbb{A}}^c\right)}_{*}^{\mathcal{DV}}$, ${\left({\mathbb{A}}_{*}^{\mathcal{DV}}\right)}^c={\left({\mathbb{A}}^c\right)}_{\mathcal{DV}}^{*}$, and also ${\left({\mathbb{A}}_{\mathcal{FL}}^{*}\right)}^c={\left({\mathbb{A}}^c\right)}_{*}^{\mathcal{FL}}$, ${\left({\mathbb{A}}_{*}^{\mathcal{FL}}\right)}^c={\left({\mathbb{A}}^c\right)}_{\mathcal{FL}}^{*}$. Thus, (1) is satisfied.

Clearly, (2) and (3) are fulfilled from the definitions.

Based on the results in (2) and (3), all the properties in (4)–(11) are ensured. □

Remark 6.

Let $\mathbb{R}$ be an equivalence relation on Ξ, and $\mathbb{A},\mathbb{B}\in P\left(\mathrm{\Xi }\right)$. Then, the following properties hold:

(1)

${\left({\mathbb{A}}_{*}\right)}^{\mathcal{DV}}⊈\mathbb{A}⊈{\left({\mathbb{A}}^{*}\right)}_{\mathcal{DV}}$ and ${\left({\mathbb{A}}_{*}\right)}^{\mathcal{FL}}⊈\mathbb{A}⊈{\left({\mathbb{A}}^{*}\right)}_{\mathcal{FL}}$,

(2)

${(\mathbb{A}\cap \mathbb{B})}_{\mathcal{DV}}^{*}⊉{\left({\mathbb{A}}^{*}\right)}_{\mathcal{DV}}\cap {\left({\mathbb{B}}^{*}\right)}_{\mathcal{DV}}$,

(3)

${(\mathbb{A}\cap \mathbb{B})}_{\mathcal{FL}}^{*}⊉{\left({\mathbb{A}}^{*}\right)}_{\mathcal{FL}}\cap {\left({\mathbb{B}}^{*}\right)}_{\mathcal{FL}}$,

(4)

${(\mathbb{A}\cap \mathbb{B})}_{*}^{\mathcal{DV}}⊉{\left({\mathbb{A}}_{*}\right)}^{\mathcal{DV}}\cap {\left({\mathbb{B}}_{*}\right)}^{\mathcal{DV}}$,

(5)

${(\mathbb{A}\cap \mathbb{B})}_{*}^{\mathcal{FL}}⊉{\left({\mathbb{A}}_{*}\right)}^{\mathcal{FL}}\cap {\left({\mathbb{B}}_{*}\right)}^{\mathcal{FL}}$,

(6)

${(\mathbb{A}\cup \mathbb{B})}_{\mathcal{DV}}^{*}⊈{\left({\mathbb{A}}^{*}\right)}_{\mathcal{DV}}\cup {\left({\mathbb{B}}^{*}\right)}_{\mathcal{DV}}$,

(7)

${(\mathbb{A}\cup \mathbb{B})}_{\mathcal{FL}}^{*}⊈{\left({\mathbb{A}}^{*}\right)}_{\mathcal{FL}}\cup {\left({\mathbb{B}}^{*}\right)}_{\mathcal{FL}}$,

(8)

${(\mathbb{A}\cup \mathbb{B})}_{*}^{\mathcal{DV}}⊈{\left({\mathbb{A}}_{*}\right)}^{\mathcal{DV}}\cup {\left({\mathbb{B}}_{*}\right)}^{\mathcal{DV}}$,

(9)

${(\mathbb{A}\cup \mathbb{B})}_{*}^{\mathcal{FL}}⊈{\left({\mathbb{A}}_{*}\right)}^{\mathcal{FL}}\cup {\left({\mathbb{B}}_{*}\right)}^{\mathcal{FL}}$.

The following example proves the results in Remark 6.

Example 9.

Let $\mathrm{\Xi }=\{\varrho ,\varsigma ,\vartheta ,\varkappa \}$, $\mathbb{A}=\{\varrho ,\vartheta \}$ and the partitioning of Ξ is given by $\mathbb{R}=\left\{\right(\varrho ,\varsigma ),(\varsigma ,\varrho ),(\vartheta ,\vartheta ),(\varkappa ,\varkappa \left)\right\}$, that is, $\mathrm{\Xi }|\mathbb{R}=\{\{\varrho ,\varsigma \},\left\{\vartheta \right\},\left\{\varkappa \right\}\}$. Then,

(1) 

associated with the floating approximation space $(\mathrm{\Xi },\mathbb{R},\mathcal{FL})$, where $\mathcal{FL}=\{\varphi ,\{\varrho \},\{\varsigma \},\{\varkappa \left\}\right\}$, we find that ${\left({\mathbb{A}}_{*}\right)}^{\mathcal{FL}}=\mathrm{\Xi },{\left({\mathbb{A}}^{*}\right)}_{\mathcal{FL}}=\left\{\vartheta \right\}$, and thus ${\left({\mathbb{A}}_{\mathbb{R}}\right)}^{\mathcal{FL}}=\mathbb{A}={\left({\mathbb{A}}^{\mathbb{R}}\right)}_{\mathcal{FL}}$ and $\mathbb{A}$ is then an exact set.

If $\mathcal{FL}=\{\varphi ,\{\varrho \},\{\varsigma \left\}\right\}$, we find that ${\left({\mathbb{A}}_{*}\right)}^{\mathcal{FL}}=\{\varrho ,\varsigma ,\vartheta \},{\left({\mathbb{A}}^{*}\right)}_{\mathcal{FL}}=\left\{\vartheta \right\}$, and thus ${\left({\mathbb{A}}_{\mathbb{R}}\right)}^{\mathcal{FL}}=\mathbb{A}={\left({\mathbb{A}}^{\mathbb{R}}\right)}_{\mathcal{FL}}$ and $\mathbb{A}$ is then an exact set.

If $\mathcal{FL}=\{\varphi ,\{\varsigma \},\{\varkappa \left\}\right\}$, we find that ${\left({\mathbb{A}}_{*}\right)}^{\mathcal{FL}}=\mathrm{\Xi },{\left({\mathbb{A}}^{*}\right)}_{\mathcal{FL}}=\{\varrho ,\varsigma ,\vartheta \}$, and thus ${\left({\mathbb{A}}_{\mathbb{R}}\right)}^{\mathcal{FL}}=\mathbb{A}$ and ${\left({\mathbb{A}}^{\mathbb{R}}\right)}_{\mathcal{FL}}=\{\varrho ,\varsigma ,\vartheta \}$, and then ${\left({\mathbb{A}}^{\mathbb{B}}\right)}_{\mathcal{FL}}=\left\{\varsigma \right\}$ and $\mathbb{A}$ is a rough set.

If $\mathcal{FL}=\{\varphi ,\{\varsigma ,\vartheta \},\{\vartheta \left\}\right\}$, we obtain that ${\left({\mathbb{A}}_{*}\right)}^{\mathcal{FL}}=\left\{\vartheta \right\},{\left({\mathbb{A}}^{*}\right)}_{\mathcal{FL}}=\{\varrho ,\varsigma \}$, and thus ${\left({\mathbb{A}}_{\mathbb{R}}\right)}^{\mathcal{FL}}=\left\{\vartheta \right\}$ and ${\left({\mathbb{A}}^{\mathbb{R}}\right)}_{\mathcal{FL}}=\{\varrho ,\varsigma ,\vartheta \}$, and then ${\left({\mathbb{A}}^{\mathbb{B}}\right)}_{\mathcal{FL}}=\{\varrho ,\varsigma \}$ and $\mathbb{A}$ is a rough set.

If $\mathcal{FL}=\{\mathbb{K}\in 2^{\mathrm{\Xi }}:\varsigma \notin \mathbb{K}\}$, we obtain that ${\left({\mathbb{A}}_{*}\right)}^{\mathcal{FL}}=\{\vartheta ,\varkappa \},{\left({\mathbb{A}}^{*}\right)}_{\mathcal{FL}}=\varphi$, and thus ${\left({\mathbb{A}}_{\mathbb{R}}\right)}^{\mathcal{FL}}=\left\{\vartheta \right\}$, ${\left({\mathbb{A}}^{\mathbb{R}}\right)}_{\mathcal{FL}}=\mathbb{A}$, and then ${\left({\mathbb{A}}^{\mathbb{B}}\right)}_{\mathcal{FL}}=\left\{\varrho \right\}$ and $\mathbb{A}$ is a rough set.

(2) 

Associated with the diving approximation space $(\mathrm{\Xi },\mathbb{R},\mathcal{DV})$, where

$\mathcal{DV}=\{\mathrm{\Xi },\{\varrho ,\varsigma \},\{\varrho ,\varsigma ,\vartheta \left\}\right\}$, we find that ${\left({\mathbb{A}}_{*}\right)}^{\mathcal{DV}}=\{\vartheta ,\varkappa \},{\left({\mathbb{A}}^{*}\right)}_{\mathcal{DV}}=\{\varrho ,\varsigma ,\vartheta \}$, and thus ${\left({\mathbb{A}}_{\mathbb{R}}\right)}^{\mathcal{DV}}=\left\{\vartheta \right\}$ and ${\left({\mathbb{A}}^{\mathbb{R}}\right)}_{\mathcal{DV}}=\{\varrho ,\varsigma ,\vartheta \}$, and then ${\left({\mathbb{A}}^{\mathbb{B}}\right)}_{\mathcal{DV}}=\{\varrho ,\varsigma \}$ and $\mathbb{A}$ is a rough set.

If $\mathcal{DV}=\{\mathrm{\Xi },\{\varrho ,\varsigma ,\varkappa \},\{\varrho ,\vartheta ,\varkappa \},\{\varsigma ,\vartheta ,\varkappa \left\}\right\}$, we find that ${\left({\mathbb{A}}_{*}\right)}^{\mathcal{DV}}=\{\varrho ,\varsigma ,\vartheta \},{\left({\mathbb{A}}^{*}\right)}_{\mathcal{DV}}=\varphi$, and thus ${\left({\mathbb{A}}_{\mathbb{R}}\right)}^{\mathcal{DV}}=\mathbb{A}$ and ${\left({\mathbb{A}}^{\mathbb{R}}\right)}_{\mathcal{DV}}=\mathbb{A}$, and hence ${\left({\mathbb{A}}^{\mathbb{B}}\right)}_{\mathcal{DV}}=\varphi$ and $\mathbb{A}$ is an exact set.

If $\mathcal{DV}=\{\mathbb{K}\in 2^{\mathrm{\Xi }}:\varrho \in \mathbb{K}\}$, we find ${\mathcal{DV}}^c=\{\mathbb{K}\in 2^{\mathrm{\Xi }}:\varrho \notin \mathbb{K}\}$ and then ${\left({\mathbb{A}}_{*}\right)}^{\mathcal{DV}}=\mathrm{\Xi },{\left({\mathbb{A}}^{*}\right)}_{\mathcal{DV}}=\{\varrho ,\varsigma \}$, and thus ${\left({\mathbb{A}}_{\mathbb{R}}\right)}^{\mathcal{DV}}=\mathbb{A}$, ${\left({\mathbb{A}}^{\mathbb{R}}\right)}_{\mathcal{DV}}=\{\varrho ,\varsigma ,\vartheta \}$, and hence ${\left({\mathbb{A}}^{\mathbb{B}}\right)}_{\mathcal{DV}}=\left\{\varsigma \right\}$ and $\mathbb{A}$ is a rough set.

If $\mathcal{DV}=\{\mathbb{K}\in 2^{\mathrm{\Xi }}:\varsigma \in \mathbb{K}\}$, we obtain ${\mathcal{DV}}^c=\{\mathbb{K}\in 2^{\mathrm{\Xi }}:\varsigma \notin \mathbb{K}\}$ and then ${\left({\mathbb{A}}_{*}\right)}^{\mathcal{DV}}=\{\vartheta ,\varkappa \},{\left({\mathbb{A}}^{*}\right)}_{\mathcal{DV}}=\varphi$, and thus ${\left({\mathbb{A}}_{\mathbb{R}}\right)}^{\mathcal{DV}}=\left\{\vartheta \right\}$, ${\left({\mathbb{A}}^{\mathbb{R}}\right)}_{\mathcal{DV}}=\mathbb{A}$, and hence ${\left({\mathbb{A}}^{\mathbb{B}}\right)}_{\mathcal{DV}}=\left\{\varrho \right\}$ and $\mathbb{A}$ is a rough set.

Lemma 2.

The lower sets and the upper sets satisfy the following properties:

(1)

${\mathbb{A}}_{\mathbb{R}}^{\mathcal{DV}}\subseteq \mathbb{A}\subseteq {\mathbb{A}}_{\mathcal{DV}}^{\mathbb{R}}$ and ${\mathbb{A}}_{\mathbb{R}}^{\mathcal{FL}}\subseteq \mathbb{A}\subseteq {\mathbb{A}}_{\mathcal{FL}}^{\mathbb{R}}$,

(2)

${\mathrm{\Xi }}_{\mathbb{R}}^{\mathcal{DV}}=\mathrm{\Xi }$, ${\varphi }_{\mathcal{DV}}^{\mathbb{R}}=\varphi$, ${\mathrm{\Xi }}_{\mathbb{R}}^{\mathcal{FL}}=\mathrm{\Xi }$ and ${\varphi }_{\mathbb{R}}^{\mathcal{FL}}={\varphi }_{\mathcal{FL}}^{\mathbb{R}}=\varphi$,

(3)

${\mathbb{A}}_{\mathcal{DV}}^{\mathbb{R}}={\left({\mathbb{A}}^c\right)}_{\mathbb{R}}^{\mathcal{DV}}{)}^c$ and ${\mathbb{A}}_{\mathcal{FL}}^{\mathbb{R}}={\left({\mathbb{A}}^c\right)}_{\mathbb{R}}^{\mathcal{FL}}{)}^c$,

(4)

$\mathbb{A}\subseteq \mathbb{B}$ implies that ${\mathbb{A}}_{\mathbb{R}}^{\mathcal{DV}}\subseteq {\mathbb{B}}_{\mathbb{R}}^{\mathcal{DV}}$, ${\mathbb{A}}_{\mathbb{R}}^{\mathcal{FL}}\subseteq {\mathbb{B}}_{\mathbb{R}}^{\mathcal{FL}}$,

(5)

$\mathbb{A}\subseteq \mathbb{B}$ implies that ${\mathbb{A}}_{\mathcal{DV}}^{\mathbb{R}}\subseteq {\mathbb{B}}_{\mathcal{DV}}^{\mathbb{R}}$, ${\mathbb{A}}_{\mathcal{FL}}^{\mathbb{R}}\subseteq {\mathbb{B}}_{\mathcal{FL}}^{\mathbb{R}}$,

(6)

${(\mathbb{A}\cap \mathbb{B})}_{\mathcal{DV}}^{\mathbb{R}}\subseteq {\mathbb{A}}_{\mathcal{DV}}^{\mathbb{R}}\cap {\mathbb{B}}_{\mathcal{DV}}^{\mathbb{R}}$,

(7)

${(\mathbb{A}\cap \mathbb{B})}_{\mathcal{FL}}^{\mathbb{R}}\subseteq {\mathbb{A}}_{\mathcal{FL}}^{\mathbb{R}}\cap {\mathbb{B}}_{\mathcal{FL}}^{\mathbb{R}}$,

(8)

${(\mathbb{A}\cap \mathbb{B})}_{\mathbb{R}}^{\mathcal{DV}}\subseteq {\mathbb{A}}_{\mathbb{R}}^{\mathcal{DV}}\cap {\mathbb{B}}_{\mathbb{R}}^{\mathcal{DV}}$,

(9)

${(\mathbb{A}\cap \mathbb{B})}_{\mathbb{R}}^{\mathcal{FL}}\subseteq {\mathbb{A}}_{\mathbb{R}}^{\mathcal{FL}}\cap {\mathbb{B}}_{\mathbb{R}}^{\mathcal{FL}}$,

(10)

${(\mathbb{A}\cup \mathbb{B})}_{\mathcal{DV}}^{\mathbb{R}}\supseteq {\mathbb{A}}_{\mathcal{DV}}^{\mathbb{R}}\cup {\mathbb{B}}_{\mathcal{DV}}^{\mathbb{R}}$,

(11)

${(\mathbb{A}\cup \mathbb{B})}_{\mathcal{FL}}^{\mathbb{R}}\supseteq {\mathbb{A}}_{\mathcal{FL}}^{\mathbb{R}}\cup {\mathbb{B}}_{\mathcal{FL}}^{\mathbb{R}}$,

(12)

${(\mathbb{A}\cup \mathbb{B})}_{\mathbb{R}}^{\mathcal{DV}}\supseteq {\mathbb{A}}_{\mathbb{R}}^{\mathcal{DV}}\cup {\mathbb{B}}_{\mathbb{R}}^{\mathcal{DV}}$,

(13)

${(\mathbb{A}\cup \mathbb{B})}_{\mathbb{R}}^{\mathcal{FL}}\supseteq {\mathbb{A}}_{\mathbb{R}}^{\mathcal{FL}}\cup {\mathbb{B}}_{\mathbb{R}}^{\mathcal{FL}}$,

(14)

${\left({\mathbb{A}}_{\mathbb{R}}^{\mathcal{DV}}\right)}_{\mathbb{R}}^{\mathcal{DV}}\subseteq \left({\mathbb{A}}_{\mathbb{R}}^{\mathcal{DV}}\right)\subseteq {\left({\mathbb{A}}_{\mathbb{R}}^{\mathcal{DV}}\right)}_{\mathcal{DV}}^{\mathbb{R}}$,

(15)

${\left({\mathbb{A}}_{\mathcal{DV}}^{\mathbb{R}}\right)}_{\mathbb{R}}^{\mathcal{DV}}\subseteq \left({\mathbb{A}}_{\mathcal{DV}}^{\mathbb{R}}\right)\subseteq {\left({\mathbb{A}}_{\mathcal{DV}}^{\mathbb{R}}\right)}_{\mathcal{DV}}^{\mathbb{R}}$.

Proof. 

From Equations (3) and (4) and Lemma 1, we easily obtain the proof of all these results. □

6. Fuzzy Diving Spaces and Fuzzy Floating Spaces

In this section, we only show that the fuzzification of these new structures could be defined.

Definition 7.

A mapping ${\mathcal{DV}}_f:I^{\mathrm{\Xi }}\to I$ is said to be a fuzzy diving structure on Ξ if it satisfies the following conditions:

(fDV1) 

${\mathcal{DV}}_f\left(\overline{1}\right)=1$,

(fDV2) 

${\mathcal{DV}}_f\left(\overline{0}\right)=0$,

(fDV3) 

${\mathcal{DV}}_f(\xi \vee \zeta )\ge {\mathcal{DV}}_f\left(\xi \right)\wedge {\mathcal{DV}}_f\left(\zeta \right)$ for all $\xi ,\zeta \in I^{\mathrm{\Xi }}$,

(fDV4) 

${\mathcal{DV}}_f(\xi \wedge \zeta )\le {\mathcal{DV}}_f\left(\xi \right)\vee {\mathcal{DV}}_f\left(\zeta \right)$ for all $\xi ,\zeta \in I^{\mathrm{\Xi }}$.

The pair $(\mathrm{\Xi },{\mathcal{DV}}_f)$ is called a fuzzy diving space.

Definition 8.

A mapping ${\mathcal{FL}}_f:I^X\to I$ is said to be a fuzzy floating structure on Ξ if it satisfies the following conditions:

(fFL1) 

${\mathcal{FL}}_f\left(\overline{0}\right)=1$,

(fFL2) 

${\mathcal{FL}}_f\left(\overline{1}\right)=0$,

(fFL3) 

${\mathcal{FL}}_f(\xi \wedge \zeta )\ge {\mathcal{FL}}_f\left(\xi \right)\wedge {\mathcal{FL}}_f\left(\zeta \right)$ for all $\xi ,\zeta \in I^{\mathrm{\Xi }}$,

(fFL4) 

${\mathcal{FL}}_f(\xi \vee \zeta )\le {\mathcal{FL}}_f\left(\xi \right)\vee {\mathcal{FL}}_f\left(\zeta \right)$ for all $\xi ,\zeta \in I^{\mathrm{\Xi }}$.

The pair $(\mathrm{\Xi },{\mathcal{FL}}_f)$ is called a fuzzy floating space.

In the following, we give samples of diving structures and floating structures on a universal set $\mathrm{\Xi }$ in the fuzzy case.

Example 10.

Fix an element $x\in \mathrm{\Xi }$; then, based on the fuzzy point $x_1$ in $I^{\mathrm{\Xi }}$, the following structures are the fuzzy diving structure and the fuzzy floating structure on Ξ, respectively.

${\mathcal{DV}}_f\left(U\right)\ge {\displaystyle \frac{1}{6}}$ if $U\left(x\right)\ge 1\forall U\in I^{\mathrm{\Xi }}$ and ${\mathcal{FL}}_f\left(V\right)\ge {\displaystyle \frac{2}{7}}$ if $V=\overline{0}$ or $V\left(x\right)\ge 1,V\ne \overline{1}$.

Clearly, ${\mathcal{DV}}_f$ and ${\mathcal{FL}}_f$ satisfy the fuzzy conditions in Definitions 7 and 8, respectively. As simple fuzzy examples, we can define both fuzzy diving structure and fuzzy floating structure as follows:

${\mathcal{DV}}_f\left(\mathbb{V}\right)=\left\{\begin{array}{cc}1\hfill & if\mathbb{V}=\overline{1}\hfill \\ 0\hfill & if\mathbb{V}=\overline{0}\hfill \\ {\displaystyle \frac{1}{4}}\hfill & if\mathbb{V}=x_1\hfill \\ {\displaystyle \frac{1}{4}}\hfill & if\mathbb{V}=(x_1\vee y_t);t\in I\hfill \\ 0\hfill & otherwise,\hfill \end{array}\right.$

${\mathcal{FL}}_f\left(\mathbb{V}\right)=\left\{\begin{array}{cc}0\hfill & if\mathbb{V}=\overline{1}\hfill \\ 1\hfill & if\mathbb{V}=\overline{0}\hfill \\ {\displaystyle \frac{2}{3}}\hfill & if\mathbb{V}=x_t;t\in I\hfill \\ {\displaystyle \frac{2}{3}}\hfill & if\mathbb{V}=y_s;s\in I\hfill \\ 0\hfill & otherwise.\hfill \end{array}\right.$

${\mathcal{DV}}_f$ and ${\mathcal{FL}}_f$ are the fuzzy diving structure and fuzzy floating structure, respectively, on a set $\mathrm{\Xi }=\{x,y,z\}$, where both are satisfying the conditions in Definitions 7 and 8. We can see that ${\mathcal{DV}}_f$ and ${\mathcal{FL}}_f$ are not acceptable to be a fuzzy topology on Ξ, and then there is no relation between fuzzy topology and these fuzzy diving and the fuzzy floating structures defined above.

7. Conclusions

In this paper, we introduced two new structures called diving structures and floating structures, and also studied some applications to characterize these concepts. Separation axioms, connectedness, and roughness are introduced in both the diving spaces and the floating spaces. In future, we will discuss these structures in detail in the fuzzy case and explore what deviations will be faced between the ordinary case and the fuzzy case. Diving and floating structures are supposed to be a cornerstone of modern mathematical theory. Their innovative nature sure will enrich the understanding of set theory, and also open doors to practical applications in computational, biological, and physical sciences. This research article lays the groundwork for future studies that will undoubtedly expand and refine these structures, solidifying their place in the broader landscape of mathematical innovation. The production of diving structures and floating structures marks a significant leap forward in mathematical sciences, providing a new lens through which to view and analyze subsets of a universal set. These structures are not merely abstract constructs; they hold profound implications for a variety of fields. From redefining rough set theory and enhancing data clustering algorithms to modeling hierarchical biological systems and optimizing computational processes, their applications are vast and diverse. The study also highlights the potential of these structures in fuzzy systems, paving the way for their integration into probabilistic models and systems with inherent uncertainties. One of the key takeaways from this research is the ability of diving and floating structures to unify and extend concepts from classical topology while maintaining their independence from it. Their flexibility in defining new forms of connectedness, separation axioms, and approximation spaces enhances their utility in fields ranging from data science to physics. Furthermore, the possibility of extending these structures to the fuzzy case offers a promising avenue for future exploration, particularly in artificial intelligence and decision-making processes.