Some Categories of Compactness and Connectedness in Generalized Topological Spaces

Abstract

The presented work enhances the study of topological characteristics in non-classical circumstances by examining $S_g^{*}$-compact and $S_g^{*}$-connected spaces in the context of generalized topological spaces. The notions of $S_g^{*}$-compactness and $S_g^{*}$-connectedness are explored better to understand their characteristics and behavior in these generalized topologies. Further, the study implements the same extended contexts to investigate the separation axioms, particularly $T_0$, $T_1$, and $T_2$. The structural consequences of these axioms, which are essential for categorizing topological spaces according to the distinctness of points and sets, are examined. Since the premises of $S_g^{*}$-compactness and $S_g^{*}$-connectedness are not directly related to the separation axioms, exploring them independently enhances the analysis of generalized topology. This study serves as theoretical insights and establishes the foundation for future research in generalized topological spaces, contributing to their continued evolution.

1. Introduction

Topology, as a cornerstone of modern mathematics, offers a framework for analyzing continuity, convergence, and separation across various spaces. Among its core concepts, compactness and connectedness stand out for their theoretical significance and practical applications. The study of generalized topological spaces has seen significant advancements through the works of various researchers. Levine’s [1] concept of generalized closed sets is important in general topology and has been thoroughly studied by many topologists in recent years. The analysis of generalized closed sets has produced intriguing new ideas, such as new covering properties and separation axioms. Császár [2] introduced the concept of generalized open sets, a foundational element that has greatly influenced subsequent studies in this area. Khayyeri and Mohamadian [3] exploration provides a critical framework for understanding the structural aspects of these spaces. Building on this, Sarsak [4] examined key properties of generalized open sets, shedding light on their behavior and further enriching the theoretical landscape of generalized topologies. These contributions are essential to the ongoing efforts to extend classical topological concepts into more generalized and versatile frameworks. The work of Maki, Balachandran and Devi [5], Maki, Rao, and Gani [6], and Navalagi and Page [7] has been instrumental in advancing the understanding of semi-generalized closed sets, generalized semi-closed sets, and their generalizations, such as semi-open and pre-open sets, which lay the foundation for further explorations into topological structures respectively. Advancements in generalized topological spaces have significantly enriched the field of topology. The concept of $S_g^{*}$-open sets was recently introduced by Missier and Jesti [8], including several additional functions in $S_g^{*}$-open sets. The idea of $\mu$-compactness in $\mathcal{GTS}$, introduced by Thomas and John [9], provided a refined understanding of compactness in a generalized framework. Extending these notions to generalized topological spaces has enabled mathematicians to address more complex structures and solve broader classes of problems.

To classify topological spaces according to the distinguishability of points and sets, this study also focuses on the separation axioms, namely $T_0$, $T_1$, and $T_2$. Urysohn [10] began the systematic exploration of separation axioms in topology. This work was later expanded by Freudenthal and Est [11] and offered a more comprehensive discussion of these concepts. In the intermediate space between $T_0$ and $T_1$, significant contributions were made by Youngs [12]. These studies laid the groundwork for further innovations, including Stone’s [13] result, which showed that any topological space can be transformed into a $T_0$ space by identifying indistinguishable points. These foundational works continue to inform and inspire contemporary research in the field.

In this context, the $S_g^{*}$-compact and $S_g^{*}$-connected spaces offer a new dimension to the study of generalized topological spaces. By examining the relationship between separation axioms such as $T_0$, $T_1$, and $T_2$, this work explores how these axioms influence the structural properties of these spaces. The study delves into the categorization of $S_g^{*}$-$T_0$, $S_g^{*}$-$T_1$, and $S_g^{*}$-$T_2$ axioms, which contribute to the overall behavior of these extended topological frameworks. This work aims to further clarify the connections between separation axioms and the unique properties of $S_g^{*}$-sets, advancing our understanding of these spaces.

2. Preliminaries

This section explains the main section by discussing various definitions and outcomes from the literature.

Definition 1

([3]). Assume a set $\mathcal{W}\ne$ ϕ. A collection ${\tau }_g$ ⊆ $2^{\mathcal{W}}$ ($2^{\mathcal{W}}$ is the power set of $\mathcal{W}$) is referred to as a generalized topology ($\mathcal{GT}$) on $\mathcal{W}$ if ϕ $\in$ and all unions of non-empty sub-classes of ${\tau }_g$ are part of ${\tau }_g$. The pair ($\mathcal{W}$, ${\tau }_g$) is known as generalized topological space ($\mathcal{GTS}$).

Remark 1

([2]). The members of ${\tau }_g$ are indicated as open. Perceive $\mathcal{E}$ to be any subset of ($\mathcal{W}$, ${\tau }_g$). $\mathcal{E}$ is considered as closed if ($\mathcal{W}$∖$\mathcal{E}$) is open. Closure of $\mathcal{E}$ is indicated by $C{l}_g$($\mathcal{E}$) is the intersection of all closed set that contains $\mathcal{E}$. The interior is represented as $In{t}_g$($\mathcal{E}$), which is the union of all open sets that are contained in $\mathcal{E}$.

Definition 2

([2]). Assume that an operator ψ:$\mathcal{W}$→$2^{2^{\mathcal{W}}}$ in $\mathcal{GTS}$ satisfies x $\in$ for $\mathcal{F}$ ψ(x). Therefore, $\mathcal{F}$ ψ(x) is referred to a generalized neighbourhood of a point x in $\mathcal{W}$. The collection of all generalized neighbourhood of $\mathcal{W}$ is termed as Ψ($\mathcal{W}$).

Definition 3

([4,5,6,7]).

1. 

Consider ($\mathcal{W}$, ${\tau }_g$) as a $\mathcal{GTS}$. Assume that $\mathcal{E}$ as a subset of ($\mathcal{W}$, ${\tau }_g$) is generalized ${\tau }_g$-semi-open if there is an open set $\mathcal{F}$ in $\mathcal{W}$ such that $\mathcal{F}\subseteq \mathcal{E}\subseteq Cl\left(\mathcal{F}\right)$ or appropriately, if $\mathcal{E}$ ⊆ Cl(Int($\mathcal{E}$)).

2. 

The complement of a generalized ${\tau }_g$-semi-open set is a generalized ${\tau }_g$-semi-closed set. SO($\mathcal{W}$, ${\tau }_g$) is the family of all generalized ${\tau }_g$-semi-open sets in ($\mathcal{W}$, ${\tau }_g$).

3. 

The union of all generalized ${\tau }_g$-semi-open sets of $\mathcal{W}$ contained in $\mathcal{E}$ is generalized semi-interior of $\mathcal{E}$ (shortly, $s_{g}Int\left(\mathcal{E}\right)$).

4. 

The intersection of all generalized semi−closed sets of $\mathcal{W}$ containing $\mathcal{E}$ is the generalized semi-closure of $\mathcal{E}$ (briefly $s_{g}Cl\left(\mathcal{E}\right)$).

Definition 4

([9]). A generalized topological space ($\mathcal{W}$, ${\tau }_g$) is called ${\tau }_g$-compact if every open cover of ($\mathcal{W}$, ${\tau }_g$) has a finite sub-cover.

Definition 5

([14]). Let ($\mathcal{W}$, ${\tau }_{g1}$) and ($\mathcal{Y}$, ${\tau }_{g1}$) be a $\mathcal{GTS}$. A mapping $\mathfrak{j}$:($\mathcal{W}$, ${\tau }_{g1}$) → ($\mathcal{Y}$, ${\tau }_{g2}$) is called ${\tau }_g$-$S_g^{*}$-irresolute if the inverse image of every ${\tau }_g$-$S_g^{*}$-open set in ($\mathcal{Y}$, ${\tau }_{g2}$) is a ${\tau }_g$-$S_g^{*}$-open in ($\mathcal{W}$, ${\tau }_{g1}$).

3. Main Results

The main section of this article systematically explores the compactness, connectedness, and separation axioms within generalized topological frameworks. This part focuses on generalized topology, where the concepts of $S_g^{*}$-compactness, $S_g^{*}$-connectedness, and separation axioms $T_0$, $T_1$, and $T_2$ are introduced, analyzed, and supported with theoretical insights and examples.

3.1. ${\tau }_g$-$S_g^{*}$-Compact Space in $\mathcal{GTS}$

Definition 6.

1. 

Consider ($\mathcal{W}$, ${\tau }_g$) to be $\mathcal{GTS}$. A subset $\mathcal{E}$ of $\mathcal{W}$ is generalized semi-generalized closed (briefly ${\tau }_g$-Sg-closed) if $s_g$Cl($\mathcal{E}$) ⊆$\mathcal{F}$ whenever $\mathcal{E}$⊆$\mathcal{F}$ and $\mathcal{F}$ is generalized ${\tau }_g$-semi-open in $\mathcal{W}$.

2. 

The complement of a generalized semi-generalized closed set is a generalized semi-generalized open. It is indicated as ${\tau }_g$-Sg-open.

3. 

The generalized semi-generalized interior (briefly $s_{g}In{t}^{*}$($\mathcal{E}$)) of $\mathcal{E}$ is specified as the union of all ${\tau }_g$-Sg-open sets of $\mathcal{W}$ contained in $\mathcal{E}$.

4. 

Assume a subset $\mathcal{E}$ of a space $\mathcal{W}$. The generalized semi-generalized closure (briefly $s_{g}C{l}^{*}$($\mathcal{E}$)) of $\mathcal{E}$ is termed as the intersection of all ${\tau }_g$-Sg-closed sets in $\mathcal{W}$ containing $\mathcal{E}$.

Definition 7.

A subset $\mathcal{E}$ of a generalized topological space ($\mathcal{W}$, ${\tau }_g$) is called generalized ${\mathrm{S}}_g^{*}$-open (${\tau }_g$-$S_g^{*}$-open shortly) if there is an open set $\mathcal{F}\in {\tau }_g$ such that $\mathcal{F}$⊆$\mathcal{E}$⊆$s_{g}C{l}^{*}$($\mathcal{F}$). The collection of all ${\tau }_g$-$S_g^{*}$-open sets in ($\mathcal{W}$, ${\tau }_g$) is represented by $S_g^{*}$O($\mathcal{W}$, ${\tau }_g$).

Definition 8.

A generalized topological space ($\mathcal{W}$, ${\tau }_g$) is called ${\tau }_g$-semi-compact if every semi-open cover of ($\mathcal{W}$, ${\tau }_g$) has a finite subcover.

Definition 9.

Assume a generalized topological space ($\mathcal{W}$, ${\tau }_g$) and a subset $\mathcal{E}$ of ($\mathcal{W}$, ${\tau }_g$). A collection $\{{\mathcal{E}}_i:i\in \wedge \}$ of ${\tau }_g$-$S_g^{*}$-open set in ($\mathcal{W}$, ${\tau }_g$) is referred to as ${\tau }_g$-$S_g^{*}$-open cover of $\mathcal{E}$ if $\mathcal{E}\subset {\cup }_{i\in \wedge }$${\mathcal{E}}_i$.

Definition 10.

If every ${\tau }_g$-$S_g^{*}$-open cover of ($\mathcal{W}$, ${\tau }_g$) has a finite subcover, then the generalized topological space ($\mathcal{W}$, ${\tau }_g$) is called ${\tau }_g$-$S_g^{*}$-compact.

Definition 11.

A subset $\mathcal{E}$ of a generalized topological space ($\mathcal{W}$, ${\tau }_g$) is called ${\tau }_g$-$S_g^{*}$-compact relative to $\mathcal{W}$ if, for every collection $\{O_i:i\in \wedge \}$ of ${\tau }_g$-$S_g^{*}$-open subsets of $\mathcal{W}$ such that $\mathcal{E}\subseteq \cup \{O_i:i\in \wedge \}$, there is a finite subset ${\wedge }_o\subseteq \wedge$ such that $\mathcal{E}\subseteq \cup \{O_i:i\in {\wedge }_o\}$.

Definition 12.

Assume $\mathcal{E}$ to be a subset of generalized topological space ($\mathcal{W}$, ${\tau }_g$). If $\mathcal{E}$ is ${\tau }_g$-$S_g^{*}$-compact as a subspace of $\mathcal{W}$, then $\mathcal{E}$ is called ${\tau }_g$-$S_g^{*}$-compact.

Theorem 1.

1. 

Every ${\tau }_g$-semi-compact space is ${\tau }_g$-$S_g^{*}$-compact.

2. 

Every ${\tau }_g$-$S_g^{*}$-compact space is ${\tau }_g$-compact.

Proof. 

1 and 2 originates from Definition 4, Definition 8, and Definition 10. □

Example 1.

Let $\mathcal{W}$ = $\mathbb{N}$ (the set of natural numbers), and let ${\tau }_g$ be the generalized topology on $\mathbb{N}$ defined as follows:

• ${\tau }_g=\{\varphi , \{$n + 2, n + 3, n + 4, …$\left\}\right|n\in \mathbb{N}\}$.
• Now, consider the subset $\mathcal{E}=\{1, 2, 3\}$. To check if $\mathcal{E}$ is ${\tau }_g$-semi-compact, let $\{O_i:i\in \wedge \}$ be a collection of ${\tau }_g$-$S_g^{*}$-open sets such that $\mathcal{E}\subseteq \cup \{O_i:i\in \wedge \}$. For any such cover, we can always find a finite subcover since $\mathcal{E}$ is finite and hence ${\tau }_g$-$S_g^{*}$-semi-compact.
• Now, let $\{O_i:i\in \wedge \}$ be a collection of ${\tau }_g$-$S_g^{*}$-open sets such that $\mathcal{E}\subseteq \cup \{O_i:i\in \wedge \}$. Since $\mathcal{E}$ is finite and every point in $\mathcal{E}$ must be covered by some $O_i$, a finite subset ${\wedge }_o\subseteq \wedge$ will suffice to cover $\mathcal{E}$. Therefore, $\mathcal{E}$ is ${\tau }_g$-$S_g^{*}$-compact.
• For $\mathcal{E}$, any open cover in the generalized topology ${\tau }_g$ will also have a finite subcover since $\mathcal{E}$ is finite. Thus, $\mathcal{E}$ is ${\tau }_g$-compact.

Theorem 2.

Every ${\tau }_g$-$S_g^{*}$-closed subset of a ${\tau }_g$-$S_g^{*}$-compact space is ${\tau }_g$-$S_g^{*}$-compact relative to ($\mathcal{W}$, ${\tau }_g$).

Proof. 

Assume that $\mathcal{E}$ as a ${\tau }_g$-$S_g^{*}$-closed subset of ${\tau }_g$-$S_g^{*}$-compact space ($\mathcal{W}$, ${\tau }_g$) ⇒ ${\mathcal{E}}^c$ is ${\tau }_g$-$S_g^{*}$-open in ($\mathcal{W}$, ${\tau }_g$). Consider {$O_i$:i ∈ ∧} be a cover of $\mathcal{E}$ by ${\tau }_g$-$S_g^{*}$-open subset of $\mathcal{W}$ such that $\mathcal{E}$ ⊂ ∪{$O_i$:i ∈ ∧} ⇒ ${\mathcal{E}}^c$ ⊂ ∪{$O_i$:i ∈ ∧} = $\mathcal{W}$. Consequently, ($\mathcal{W}$, ${\tau }_g$) is ${\tau }_g$-$S_g^{*}$-compact, so there is a finite subset ${\mathcal{E}}_o$ of $\mathcal{E}$ such that $\mathcal{E}$ ⊂ ${\mathcal{E}}^c$ ∪ {$O_i$:i ∈ ∧} = $\mathcal{W}$. Then $\mathcal{E}$ ⊂ ∪ {$O_i$:i ∈ ∧} and therefore $\mathcal{E}$ is ${\tau }_g$-$S_g^{*}$-compact relative to $\mathcal{W}$. □

Theorem 3.

Assume a surjective ${\tau }_g$-$S_g^{*}$-continuous map $\mathfrak{h}$: ($\mathcal{W}$, ${\tau }_{g1}$) → ($\mathcal{Y}$, ${\tau }_{g2}$). If ($\mathcal{W}$, ${\tau }_{g1}$) is ${\tau }_g$-$S_g^{*}$-compact, then ($\mathcal{Y}$, ${\tau }_{g2}$) is ${\tau }_g$-compact.

Proof. 

Consider an open cover {${\mathcal{E}}_i$:i ∈ ∧} of $\mathcal{Y}$. $\mathfrak{h}$ is ${\tau }_g$-$S_g^{*}$-continuous ⇒ {${\mathfrak{h}}^{-1}$(${\mathcal{E}}_i$):i ∈ ∧} is a ${\tau }_g$-$S_g^{*}$-open cover of $\mathcal{W}$. Furthermore, there is a finite ${\tau }_g$-subcover {${\mathfrak{h}}^{-1}$(${\mathcal{E}}_1$), ${\mathfrak{h}}^{-1}$(${\mathcal{E}}_2$), ${\mathfrak{h}}^{-1}$(${\mathcal{E}}_3$), …, ${\mathfrak{h}}^{-1}$(${\mathcal{E}}_n$)}, as $\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-compact. The surjectiveness of $\mathcal{W}$ ⇒ {${\mathcal{E}}_1$, ${\mathcal{E}}_2$, ${\mathcal{E}}_3$, …, ${\mathcal{E}}_n$} is a finite ${\tau }_g$-subcover of $\mathcal{Y}$ ⇒ $\mathcal{Y}$ is ${\tau }_g$-compact. □

Theorem 4.

Assume a surjective ${\tau }_g$-$S_g^{*}$-irresolute map $\mathfrak{h}$: ($\mathcal{W}$, ${\tau }_{g1}$) → ($\mathcal{Y}$, ${\tau }_{g2}$). If ($\mathcal{W}$, ${\tau }_{g1}$) is ${\tau }_g$-$S_g^{*}$-compact, then ($\mathcal{Y}$, ${\tau }_{g2}$) is ${\tau }_g$-$S_g^{*}$-compact.

Proof. 

Consider a ${\tau }_g$-$S_g^{*}$-open cover {${\mathcal{E}}_i$:i ∈ ∧} of $\mathcal{Y}$. $\mathfrak{h}$ is ${\tau }_g$-$S_g^{*}$-irresolute ⇒ {${\mathfrak{h}}^{-1}$ (${\mathcal{E}}_i$):i ∈ ∧} is a ${\tau }_g$-$S_g^{*}$-open cover of $\mathcal{W}$. Furthermore, there is a finite ${\tau }_g$-subcover {${\mathfrak{h}}^{-1}$(${\mathcal{E}}_1$), ${\mathfrak{h}}^{-1}$(${\mathcal{E}}_2$), ${\mathfrak{h}}^{-1}$(${\mathcal{E}}_3$), …, ${\mathfrak{h}}^{-1}$(${\mathcal{E}}_n$)}, as $\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-compact. Now $\mathfrak{h}$ implies that {${\mathcal{E}}_1$, ${\mathcal{E}}_2$, ${\mathcal{E}}_3$, …, ${\mathcal{E}}_n$} is a finite ${\tau }_g$-subcover of $\mathcal{Y}$ ⇒ $\mathcal{Y}$ is ${\tau }_g$-$S_g^{*}$-compact. □

Theorem 5.

If a ${\tau }_g$-$S_g^{*}$-irresolute map $\mathfrak{h}$: ($\mathcal{W}$, ${\tau }_{g1}$) → ($\mathcal{Y}$, ${\tau }_{g2}$) and a subset $\mathcal{D}$ of ($\mathcal{W}$, ${\tau }_{g1}$) is ${\tau }_g$-$S_g^{*}$-compact relative to $\mathcal{W}$, then the image $\mathfrak{h}$($\mathcal{D}$) is ${\tau }_g$-$S_g^{*}$-compact relative to $\mathcal{Y}$.

Proof. 

Let {${\mathcal{E}}_i$:i ∈ ∧} be any collection of ${\tau }_g$-$S_g^{*}$-open subsets of $\mathcal{Y}$ such that $\mathfrak{h}$($\mathcal{D}$) ⊂ ∪ {${\mathcal{E}}_i$:i ∈ ∧}. Then $\mathcal{D}$ ⊂ ∪ {${\mathfrak{h}}^{-1}$(${\mathcal{E}}_i$):i ∈ ∧} holds. $\mathcal{D}$ is ${\tau }_g$-$S_g^{*}$-compact relative to $\mathcal{W}$ by hypothesis, so there is a ${\wedge }_o$, a finite subset of ∧, such that $\mathcal{D}$ ⊂ ∪ {${\mathfrak{h}}^{-1}$(${\mathcal{E}}_i$):i $\in {\wedge }_o$} ⇒ $\mathfrak{h}$($\mathcal{D}$) ⊂ ∪ {${\mathcal{E}}_i$:i $\in {\wedge }_o$}. Therefore, $\mathfrak{h}$($\mathcal{D}$) is ${\tau }_g$-$S_g^{*}$-compact relative to $\mathcal{Y}$. □

Theorem 6.

If a surjective map $\mathfrak{h}$: ($\mathcal{W}$, ${\tau }_{g1}$) → ($\mathcal{Y}$, ${\tau }_{g2}$) is strongly ${\tau }_g$-$S_g^{*}$-continuous, and ($\mathcal{W}$, ${\tau }_{g1}$) is ${\tau }_g$-compact, then ($\mathcal{Y}$, ${\tau }_{g2}$) is ${\tau }_g$-$S_g^{*}$-compact.

Proof. 

Consider a ${\tau }_g$-$S_g^{*}$-open cover {${\mathcal{E}}_i$:i ∈ ∧} of $\mathcal{Y}$. $\mathfrak{h}$ is strongly ${\tau }_g$-$S_g^{*}$-continuous ⇒{${\mathfrak{h}}^{-1}$ (${\mathcal{E}}_i$):i ∈ ∧} is an open cover of $\mathcal{W}$. Furthermore, there is a finite ${\tau }_g$-subcover {${\mathfrak{h}}^{-1}$(${\mathcal{E}}_1$), ${\mathfrak{h}}^{-1}$(${\mathcal{E}}_2$), ${\mathfrak{h}}^{-1}$(${\mathcal{E}}_3$), …, ${\mathfrak{h}}^{-1}$(${\mathcal{E}}_n$)}, as $\mathcal{W}$ is ${\tau }_g$-compact. The surjectiveness of $\mathfrak{h}$⇒{${\mathcal{E}}_1$, ${\mathcal{E}}_2$, ${\mathcal{E}}_3$, …, ${\mathcal{E}}_n$} is a finite subcover of $\mathcal{Y}$ ⇒ $\mathcal{Y}$ is ${\tau }_g$-$S_g^{*}$-compact. □

Theorem 7.

If a surjective map $\mathfrak{h}$: ($\mathcal{W}$, ${\tau }_{g1}$) → ($\mathcal{Y}$, ${\tau }_{g2}$) is perfectly ${\tau }_g$-$S_g^{*}$-continuous and ($\mathcal{W}$, ${\tau }_{g1}$) is ${\tau }_g$-compact, then ($\mathcal{Y}$, ${\tau }_{g2}$) is ${\tau }_g$-$S_g^{*}$-compact.

Proof. 

Every perfectly ${\tau }_g$-$S_g^{*}$-continuous is strongly ${\tau }_g$-$S_g^{*}$-continuous. Consequently, the result is derived from Theorem 6. □

3.2. ${\tau }_g$-$S_g^{*}$-Connected Space

Definition 13.

Assume a generalized topological space ($\mathcal{W}$, ${\tau }_g$) and two disjoint non-empty open sets $\mathcal{D}$ and $\mathcal{E}$ in $\mathcal{W}$. Then ($\mathcal{W}$, ${\tau }_g$) is known as ${\tau }_g$-disconnection if $\mathcal{D}$ ∪ $\mathcal{E}$ = $\mathcal{W}$.

Definition 14.

Consider a generalized topological space ($\mathcal{W}$, ${\tau }_g$) and two disjoint non-empty open sets $\mathcal{D}$ and $\mathcal{E}$ in $\mathcal{W}$. Then ($\mathcal{W}$, ${\tau }_g$) is known as a ${\tau }_g$-connected space if ($\mathcal{W}$, ${\tau }_g$) has no ${\tau }_g$-disconnection {$\mathcal{D}$, $\mathcal{E}$}.

Definition 15.

Assume a generalized topological space ($\mathcal{W}$, ${\tau }_g$) and two disjoint non-empty ${\tau }_g$-$S_g^{*}$-open sets $\mathcal{D}$, and $\mathcal{E}$ in ($\mathcal{W}$, ${\tau }_g$) does not exist as $\mathcal{D}$ ∪ $\mathcal{E}$ = $\mathcal{W}$. Then space ($\mathcal{W}$, ${\tau }_g$) is known as ${\tau }_g$-$S_g^{*}$-connected space, otherwise called ${\tau }_g$-$S_g^{*}$-disconnected space if $\mathcal{D}$ ∪ $\mathcal{E}$ = $\mathcal{W}$.

Example 2.

Consider $\mathcal{W}$ = $\{e_1,f_1,d_1\}$ and ${\tau }_g$ = $\{\varphi , \{f_1\}$, $\left\{d_1\right\}$, $\{f_1,d_1\}\}$. In this space ($\mathcal{W}$, ${\tau }_g$)$S_g^{*}O$ $\{\varphi , \{f_1\}$, $\left\{d_1\right\}$, $\{f_1,d_1\}\}$. Thus, there does not exist any non-empty ${\tau }_g$-$S_g^{*}$-open sets $\mathcal{D}$ and $\mathcal{E}$ such that $\mathcal{D}\cup \mathcal{E}=\mathcal{W}$ ⇒ ($\mathcal{W}$, ${\tau }_g$) is ${\tau }_g$-$S_g^{*}$-connected space.

Theorem 8.

Every ${\tau }_g$-$S_g^{*}$-connected space is ${\tau }_g$-connected space.

Proof. 

Let ($\mathcal{W}$, ${\tau }_g$) be a ${\tau }_g$-$S_g^{*}$-connected space. Assume ($\mathcal{W}$, ${\tau }_g$) is not a ${\tau }_g$-connected space. Then for any two disjoint non-empty open subsets $\mathcal{D}$ and $\mathcal{E}$ in $\mathcal{W}$, $\mathcal{D}$ ∪ $\mathcal{E}$ = $\mathcal{W}$. Every open set is ${\tau }_g$-$S_g^{*}$-open set. Thus, $\mathcal{D}$ and $\mathcal{E}$ are also ${\tau }_g$-$S_g^{*}$-open sets, and $\mathcal{D}$ ∪ $\mathcal{E}$ = $\mathcal{W}$. Then ($\mathcal{W}$, ${\tau }_g$) is not ${\tau }_g$-$S_g^{*}$-connected space, which implies a contradiction. Therefore, ($\mathcal{W}$, ${\tau }_g$) is a ${\tau }_g$-connected space. □

Remark 2.

The converse of the above theorem is invalid as demonstrated by the following illustration.

Example 3.

Consider that $\mathcal{W}$ = $\{e_1,f_1,d_1,k_1\}$ and ${\tau }_g$ = $\{\varphi , \mathcal{W}$, $\left\{e_1\right\}$, $\{f_1,d_1\}$, $\{e_1,f_1,d_1\}\}$. In ($\mathcal{W}$, ${\tau }_g$)O = $\{\varphi , \mathcal{W}$, $\left\{e_1\right\}$, $\{f_1,d_1\}$, $\{e_1,f_1,d_1\}\}$ and ($\mathcal{W}$, ${\tau }_g$)$S_g^{*}$ O = $\{\varphi , \mathcal{W}$, $\left\{e_1\right\}$, $\{f_1,d_1\}$, $\{e_1,f_1,d_1\}$, $\{f_1,d_1,k_1\}\}$, there does not exist any non-empty open sets $\mathcal{D}$ and $\mathcal{E}$ such that $\mathcal{D}\cup \mathcal{E}$ = $\mathcal{W}$ ⇒ ($\mathcal{W}$, ${\tau }_g$) is a ${\tau }_g$-connected space. However, there are two non-empty disjoint ${\tau }_g$-$S_g^{*}$-open sets $\left\{e_1\right\}$ and $\{f_1,d_1,k_1\}$ such that $\left\{e_1\right\}\cup \{f_1,d_1,k_1\}$ = $\mathcal{W}$ ⇒ ($\mathcal{W}$, ${\tau }_g$) is not a ${\tau }_g$-$S_g^{*}$-connected space.

Theorem 9.

Assume a generalized topological space ($\mathcal{W}$, ${\tau }_g$). Then the subsequent is equivalent.

i.

$\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-connected.

ii.

$\varphi$ and $\mathcal{W}$ are only a ${\tau }_g$-$S_g^{*}$-open set and a ${\tau }_g$-$S_g^{*}$-closed set in $\mathcal{W}$.

iii.

Any ${\tau }_g$-$S_g^{*}$-continuous mapping $\mathfrak{h}$: ($\mathcal{W}$, ${\tau }_{g1}$) → ($\mathcal{Y}$, ${\tau }_{g2}$) is a constant map where ($\mathcal{Y}$, ${\tau }_{g2}$) is at least a two-point discrete space.

Proof. 

(i) then (ii): A ${\tau }_g$-$S_g^{*}$-open set and ${\tau }_g$-$S_g^{*}$-closed set $\mathcal{E}$ in ($\mathcal{W}$, ${\tau }_g$) ⇒ ${\mathcal{E}}^c$ is also a ${\tau }_g$-$S_g^{*}$-open set and a ${\tau }_g$-$S_g^{*}$-closed set in ($\mathcal{W}$, ${\tau }_g$). Then $\mathcal{E}$ ∪ ${\mathcal{E}}^c$ = $\mathcal{W}$, as $\mathcal{E}$ and ${\mathcal{E}}^c$ are both disjoint ${\tau }_g$-$S_g^{*}$-open sets, which implies a contradiction. ($\mathcal{W}$, ${\tau }_g$) is a ${\tau }_g$-$S_g^{*}$-connected space, so $\mathcal{W}$ is $\varphi$ or $\mathcal{W}$.

(ii) then (i): Consider two disjoint ${\tau }_g$-$S_g^{*}$-open sets $\mathcal{E}$ and $\mathcal{D}$ in ($\mathcal{W}$, ${\tau }_g$) and $\mathcal{E}$ ∪ $\mathcal{D}$ = $\mathcal{W}$. Since ${\mathcal{E}}^c$ = $\mathcal{D}$, $\mathcal{E}$ is a ${\tau }_g$-$S_g^{*}$-open set and a ${\tau }_g$-$S_g^{*}$-closed set. By hypothesis, $\mathcal{E}$ is $\varphi$ or $\mathcal{W}$, which is a contradiction. Hence, ($\mathcal{W}$, ${\tau }_g$) is a ${\tau }_g$-$S_g^{*}$-connected space.

(ii) then (iii): Assume a ${\tau }_g$-$S_g^{*}$-continuous mapping $\mathfrak{h}$: ($\mathcal{W}$, ${\tau }_{g1}$) → ($\mathcal{Y}$, ${\tau }_{g2}$) is a constant map where ($\mathcal{Y}$, ${\tau }_{g2}$) is at least a two-point discrete space. Then ${\mathfrak{h}}^{-1}$ ({x}) is a ${\tau }_g$-$S_g^{*}$-open and a ${\tau }_g$-$S_g^{*}$-closed set for all elements of $\mathcal{Y}$. $\mathcal{W}$ = ∪ ${\mathfrak{h}}^{-1}$ ({x}), where x $\in \mathcal{Y}$ and ${\mathfrak{h}}^{-1}$ ({x}) is ${\tau }_g$-$S_g^{*}$-open and ${\tau }_g$-$S_g^{*}$-closed in $\mathcal{W}$. By hypothesis, ${\mathfrak{h}}^{-1}$ ({x}) is $\varphi$ or $\mathcal{W}$, ∀ x ∈ $\mathcal{Y}$, then $\mathfrak{h}$ fails to map. There is, then, at least one point for x elements of $\mathcal{Y}$ where ${\mathfrak{h}}^{-1}$ ({x}) is not empty ⇒${\mathfrak{h}}^{-1}$ ({x}) = $\mathcal{W}$ ⇒ $\mathfrak{h}$ is a constant function.

(iii) then (ii): Let $\mathcal{E}$ be ${\tau }_g$-$S_g^{*}$-open and ${\tau }_g$-$S_g^{*}$-closed in a generalized topological space ($\mathcal{W}$, ${\tau }_g$) where $\mathcal{E}$ is a non-empty set. Consider a ${\tau }_g$-$S_g^{*}$-continuous mapping $\mathfrak{h}$: ($\mathcal{W}$, ${\tau }_{g1}$) → ($\mathcal{Y}$, ${\tau }_{g2}$) defined as $\mathfrak{h}$ ($\mathcal{E}$) = {x} and $\mathfrak{h}$ (${\mathcal{E}}^c$) = {y}, where x, y ∈ $\mathcal{Y}$ and x ≠ y. By hypothesis, $\mathfrak{h}$ is a constant map and $\mathcal{E}$ = $\mathcal{W}$. □

Theorem 10.

Assume $\mathfrak{h}$: ($\mathcal{W}$, ${\tau }_{g1}$) → ($\mathcal{Y}$, ${\tau }_{g2}$) is a surjective, ${\tau }_g$-$S_g^{*}$-continuous mapping and ($\mathcal{W}$, ${\tau }_{g1}$) is a ${\tau }_g$-$S_g^{*}$-connected space. Then ($\mathcal{Y}$, ${\tau }_{g2}$) is a ${\tau }_g$-connected space.

Proof. 

Assume ($\mathcal{Y}$, ${\tau }_{g2}$) is not a ${\tau }_g$-connected space. Consider two non-empty disjoint open sets $\mathcal{E}$ and $\mathcal{D}$ in $\mathcal{Y}$. For example, $\mathcal{E}$ ∪ $\mathcal{D}$ = $\mathcal{Y}$. Because $\mathfrak{h}$ is ${\tau }_g$-$S_g^{*}$-continuous, ${\mathfrak{h}}^{-1}$($\mathcal{E}$) ∪ ${\mathfrak{h}}^{-1}$($\mathcal{D}$) = $\mathcal{W}$ and ${\mathfrak{h}}^{-1}$($\mathcal{E}$). ${\mathfrak{h}}^{-1}$($\mathcal{D}$) is a ${\tau }_g$-$S_g^{*}$-open disjoint set in $\mathcal{W}$, which implies a contradiction. Thus, $\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-connected. Hence, $\mathcal{Y}$ is a ${\tau }_g$-connected space. □

Theorem 11.

Consider $\mathfrak{h}$: ($\mathcal{W}$, ${\tau }_{g1}$) → ($\mathcal{Y}$, ${\tau }_{g2}$) is a surjective, ${\tau }_g$-$S_g^{*}$-irresolute function, and ($\mathcal{W}$, ${\tau }_{g1}$) is a ${\tau }_g$-$S_g^{*}$-connected space, so ($\mathcal{Y}$, ${\tau }_{g2}$) is a ${\tau }_g$-connected space.

Proof. 

Assume ($\mathcal{Y}$, ${\tau }_{g2}$) is a ${\tau }_g$-$S_g^{*}$-connected space. Consider two non-empty ${\tau }_g$-$S_g^{*}$-open sets $\mathcal{E}$ and $\mathcal{D}$ in $\mathcal{Y}$, such as $\mathcal{E}$ ∪ $\mathcal{D}$ = $\mathcal{Y}$. $\mathfrak{h}$ is surjective, ${\tau }_g$-$S_g^{*}$-continuous, so ${\mathfrak{h}}^{-1}$($\mathcal{E}$) ∪ ${\mathfrak{h}}^{-1}$($\mathcal{D}$) = $\mathcal{W}$ and ${\mathfrak{h}}^{-1}$($\mathcal{E}$). ${\mathfrak{h}}^{-1}$($\mathcal{D}$) is a ${\tau }_g$-$S_g^{*}$-open disjoint subset in $\mathcal{W}$, which implies a contradiction. Thus, $\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-connected. Hence, $\mathcal{Y}$ is also a ${\tau }_g$-$S_g^{*}$-connected space. □

Theorem 12.

Consider $\mathfrak{h}$: ($\mathcal{W}$, ${\tau }_{g1}$) → ($\mathcal{Y}$, ${\tau }_{g2}$) is a strongly ${\tau }_g$-$S_g^{*}$-continuous function, and ($\mathcal{W}$, ${\tau }_{g1}$) is a ${\tau }_g$-connected space. Then its image is a ${\tau }_g$-$S_g^{*}$-connected space.

Proof. 

Consider $\mathfrak{h}$: ($\mathcal{W}$, ${\tau }_{g1}$) → ($\mathcal{Y}$, ${\tau }_{g2}$) is a strongly ${\tau }_g$-$S_g^{*}$-continuous map, and ($\mathcal{W}$, ${\tau }_{g1}$) is a ${\tau }_g$-connected space. Assume ($\mathcal{Y}$, ${\tau }_{g2}$) is not a ${\tau }_g$-$S_g^{*}$-connected space for two non-empty disjoint ${\tau }_g$-$S_g^{*}$-open sets $\mathcal{E}$ and $\mathcal{D}$ in $\mathcal{Y}$, such as $\mathcal{E}$ ∪ $\mathcal{D}$ = $\mathcal{Y}$. $\mathfrak{h}$ is strongly ${\tau }_g$-$S_g^{*}$-continuous, so ${\mathfrak{h}}^{-1}$($\mathcal{E}$) ∪ ${\mathfrak{h}}^{-1}$($\mathcal{D}$) = $\mathcal{W}$ and ${\mathfrak{h}}^{-1}$($\mathcal{E}$), ${\mathfrak{h}}^{-1}$($\mathcal{D}$) are open disjoint sets in $\mathcal{W}$, which implies a contradiction. Thus, $\mathcal{W}$ is ${\tau }_g$-connected; hence, $\mathcal{Y}$ is a ${\tau }_g$-$S_g^{*}$-connected space. □

3.3. ${\tau }_g$-$S_g^{*}$-Separation Axioms

Definition 16.

A generalized topological space ($\mathcal{W}$, ${\tau }_g$) is called a ${\tau }_g$-$S_g^{*}$Tc space if every ${\tau }_g$-$S_g^{*}$-closed set is closed.

Definition 17.

Consider that $\mathcal{E}$, to be a subset of $\mathcal{W}$, is said to be a ${\tau }_g$-$T_0$ space if for every explicit point $\mathfrak{r}$ and $\mathfrak{s}$ of $\mathcal{W}$, $\mathfrak{s}\notin {\mathcal{M}}_1$, $\mathfrak{r}\in {\mathcal{M}}_1$ or $\mathfrak{r}\notin {\mathcal{M}}_1$, $\mathfrak{s}\in {\mathcal{M}}_1$, where ${\mathcal{M}}_1$ is an open set of $\mathcal{W}$.

Definition 18.

Let $\mathcal{E}\subseteq \mathcal{W}$. $\mathcal{E}$ is said to be a ${\tau }_g$-$T_1$ space if for every explicit point $\mathfrak{r}$ and $\mathfrak{s}$ of $\mathcal{W}$, $\mathfrak{s}\notin {\mathcal{M}}_1$, $\mathfrak{r}\in {\mathcal{M}}_1$ and $\mathfrak{r}\notin {\mathcal{N}}_1$, $\mathfrak{s}$, where ${\mathcal{M}}_1$ and ${\mathcal{N}}_1$ are open sets of $\mathcal{W}$.

Definition 19.

Assume $\mathcal{E}\subseteq \mathcal{W}$. $\mathcal{E}$ is said to be a ${\tau }_g$-$T_2$ space if for every explicit point $\mathfrak{r}$ and $\mathfrak{s}$ of $\mathcal{W}$, $\mathfrak{s}\notin {\mathcal{M}}_1$, $\mathfrak{r}\in {\mathcal{M}}_1$ and $\mathfrak{r}\notin {\mathcal{N}}_1$, $\mathfrak{s}\in {\mathcal{N}}_1$, where ${\mathcal{M}}_1$ and ${\mathcal{N}}_1$ are open sets of $\mathcal{W}$ and ${\mathcal{M}}_1\ne {\mathcal{N}}_1$.

Definition 20.

The function $\mathfrak{j}$:$\mathcal{W}$ → $\mathcal{Y}$ is indicated as ${\tau }_g$-$S_g^{*}$-continuous if there is an inverse image that is ${\tau }_g$-$S_g^{*}$-open in ($\mathcal{W}$, ${\tau }_{g1}$) for every open set in ($\mathcal{Y}$, ${\tau }_{g2}$).

3.3.1. ${\tau }_g$-$S_g^{*}$-$T_0$, ${\tau }_g$-$S_g^{*}$-$T_1$, ${\tau }_g$-$S_g^{*}$-$T_2$ Spaces

Definition 21.

Consider that $\mathcal{E}$, to be a subset of $\mathcal{W}$, is said to be a ${\tau }_g$-$S_g^{*}$-$T_0$ space if for every explicit point $\mathfrak{r}$ and $\mathfrak{s}$ of $\mathcal{W}$, $\mathfrak{s}\notin {\mathcal{M}}_1$, $\mathfrak{r}\in {\mathcal{M}}_1$ or $\mathfrak{r}\notin {\mathcal{M}}_1$, $\mathfrak{s}\in {\mathcal{M}}_1$, where ${\mathcal{M}}_1$ is a ${\tau }_g$-$S_g^{*}$-open set of $\mathcal{W}$.

Definition 22.

Let $\mathcal{E}\subseteq \mathcal{W}$. $\mathcal{E}$ is said to be a ${\tau }_g$-$S_g^{*}$-$T_1$ space if for every explicit point $\mathfrak{r}$ and $\mathfrak{s}$ of $\mathcal{W}$, $\mathfrak{s}\notin {\mathcal{M}}_1$, $\mathfrak{r}\in {\mathcal{M}}_1$ and $\mathfrak{r}\notin {\mathcal{N}}_1$, $\mathfrak{s}\in {\mathcal{N}}_1$, where ${\mathcal{M}}_1$ and ${\mathcal{N}}_1$ are ${\tau }_g$-$S_g^{*}$-open sets of $\mathcal{W}$.

Definition 23.

Assume $\mathcal{E}\subseteq \mathcal{W}$. $\mathcal{E}$ is said to be a ${\tau }_g$-$S_g^{*}$-$T_2$(${\tau }_g$-$S_g^{*}$-Housdorff) space if for every explicit point $\mathfrak{r}$ and $\mathfrak{s}$ of $\mathcal{W}$, $\mathfrak{s}\notin {\mathcal{M}}_1$, $\mathfrak{r}\in {\mathcal{M}}_1$ and $\mathfrak{r}\notin {\mathcal{N}}_1$, $\mathfrak{s}\in {\mathcal{N}}_1$, where ${\mathcal{M}}_1$ and ${\mathcal{N}}_1$ are ${\tau }_g$-$S_g^{*}$-open sets of $\mathcal{W}$ and ${\mathcal{M}}_1\ne {\mathcal{N}}_1$.

Theorem 13.

i. 

If $\mathcal{W}$ is ${\tau }_g$-$T_0$, then $\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-$T_0$.

ii. 

If $\mathcal{W}$ is ${\tau }_g$-$T_1$, then $\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-$T_0$ and ${\tau }_g$-$S_g^{*}$-$T_1$.

iii. 

If $\mathcal{W}$ is ${\tau }_g$-$T_2$, then $\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-$T_2$.

iv. 

If $\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-$T_2$, then $\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-$T_0$.

v. 

If $\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-$T_2$, then $\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-$T_1$.

Proof. 

(i) Assume $\mathcal{W}$ is a ${\tau }_g$-$T_0$ space. There is an open set $\mathcal{F}$ for distinct points of every pair $\mathfrak{r}$ $\mathfrak{s}$ of $\mathcal{W}$ where $\mathfrak{s}$ ∉ $\mathcal{F}$, $\mathfrak{r}\in \mathcal{F}$, or $\mathfrak{r}$ ∉ $\mathcal{F}$, $\mathfrak{s}\in \mathcal{F}$. Then $\mathcal{F}\in {\tau }_g$-$S_g^{*}$O($\mathcal{W}$) with $\mathfrak{s}$ ∉ $\mathcal{F}$, $\mathfrak{r}\in \mathcal{F}$, or $\mathfrak{r}$ ∉ $\mathcal{F}$, $\mathfrak{s}\in \mathcal{F}$. Therefore, $\mathcal{W}$ is a ${\tau }_g$-$S_g^{*}$-$T_0$ space. In the same way, we can demonstrate (ii), (iii), (iv), and (v). □

See the below Figure 1 for the relation between the above theorems.

Figure 1. An illustration of the above theorems and considerations.

Remark 3.

The illustration below demonstrates that the converse of the above theorem is invalid.

Example 4.

Let $\mathcal{W}$ = $\{e_1,f_1,d_1\}$ and ${\tau }_g$ = $\{\varphi , \mathcal{W}$, $\left\{d_1\right\}$, $\{e_1,f_1\}\}$. In ($\mathcal{W}$, ${\tau }_g$)O = $\{\varphi , \mathcal{W}$, $\left\{d_1\right\}$, $\{e_1,f_1\}\}$ and ${\tau }_g$-$S_g^{*}$O = P($\mathcal{W}$). Hence, ($\mathcal{W}$, ${\tau }_g$) is as follows:

○

${\tau }_g$-$S_g^{*}$-$T_0$ but not ${\tau }_g$-$T_0$. For explicit points $\mathfrak{r}$ and $\mathfrak{s}$ of $\mathcal{W}$, no open set exists with $\mathfrak{s}\notin {\mathcal{M}}_1$, $\mathfrak{r}\in {\mathcal{M}}_1$ or $\mathfrak{r}\notin {\mathcal{M}}_1$, $\mathfrak{s}\in {\mathcal{M}}_1$, where ${\mathcal{M}}_1$ is an open set of $\mathcal{W}$.

○

${\tau }_g$-$S_g^{*}$-$T_1$ space but not ${\tau }_g$-$T_1$ space. For explicit points $\mathfrak{r}$ and $\mathfrak{s}$ of $\mathcal{W}$, no two open sets ${\mathcal{M}}_1$ and ${\mathcal{N}}_1$ exists with $\mathfrak{s}\notin {\mathcal{M}}_1$, $\mathfrak{r}\in {\mathcal{M}}_1$ and $\mathfrak{r}\notin {\mathcal{N}}_1$, $\mathfrak{s}\in {\mathcal{N}}_1$.

○

${\tau }_g$-$S_g^{*}$-$T_2$ space but not ${\tau }_g$-$T_2$ space. For explicit points $\mathfrak{r}$ and $\mathfrak{s}$ of $\mathcal{W}$, no two distinct open sets ${\mathcal{M}}_1$ and ${\mathcal{N}}_1$ exists with $\mathfrak{s}\notin {\mathcal{M}}_1$, $\mathfrak{r}\in {\mathcal{M}}_1$ and $\mathfrak{r}\notin {\mathcal{N}}_1$, $\mathfrak{s}\in {\mathcal{N}}_1$.

Theorem 14.

If $\mathfrak{h}$ is bijective and strongly ${\tau }_g$-$S_g^{*}$-open, and $\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-$T_0$, then $\mathcal{Y}$ is a ${\tau }_g$-$S_g^{*}$-$T_0$ space.

Proof. 

Take ${\mathfrak{r}}_{\mathfrak{2}}$ and ${\mathfrak{s}}_{\mathfrak{2}}$ of $\mathcal{Y}$ with ${\mathfrak{r}}_{\mathfrak{2}}\ne {\mathfrak{s}}_{\mathfrak{2}}$. By hypothesis, ${\mathfrak{r}}_{\mathfrak{2}}$ = $\mathfrak{h}$(${\mathfrak{r}}_{\mathfrak{1}}$) and ${\mathfrak{s}}_{\mathfrak{2}}$ = $\mathfrak{h}$(${\mathfrak{s}}_{\mathfrak{1}}$) where ${\mathfrak{r}}_{\mathfrak{1}}$ and ${\mathfrak{s}}_{\mathfrak{1}}$ are the explicit points of $\mathcal{W}$. By hypothesis, ${\mathcal{M}}_1\in {\tau }_g$-$S_g^{*}$O($\mathcal{W}$) with ${\mathfrak{r}}_{\mathfrak{1}}\in {\mathcal{M}}_1$ and ${\mathfrak{s}}_{\mathfrak{1}}\notin {\mathcal{M}}_1$. Therefore, $\mathfrak{h}$(${\mathfrak{r}}_{\mathfrak{1}}$) ∈ $\mathfrak{h}$(${\mathcal{M}}_1$) and $\mathfrak{h}$(${\mathfrak{s}}_{\mathfrak{1}}$) ∉ $\mathfrak{h}$(${\mathcal{M}}_1$). $\mathfrak{h}$(${\mathcal{M}}_1$) ∈ ${\tau }_g$-$S_g^{*}$O($\mathcal{Y}$) as $\mathcal{W}$ is strongly ${\tau }_g$-$S_g^{*}$-open. Thus, $\mathfrak{h}$(${\mathcal{M}}_1$) is a ${\tau }_g$-$S_g^{*}$-open set in $\mathcal{Y}$ with ${\mathfrak{r}}_{\mathfrak{2}}\in \mathfrak{h}$(${\mathcal{M}}_1$) and ${\mathfrak{s}}_{\mathfrak{2}}$ ∉ $\mathfrak{h}$(${\mathcal{M}}_1$). Therefore, $\mathcal{Y}$ is a ${\tau }_g$-$S_g^{*}$-$T_0$ space. □

Theorem 15.

If $\mathfrak{h}$ is injective and ${\tau }_g$-$S_g^{*}$-irresolute, and $\mathcal{Y}$ is ${\tau }_g$-$S_g^{*}$-$T_0$, then $\mathcal{X}$ is a ${\tau }_g$-$S_g^{*}$-$T_0$ space.

Proof. 

Take ${\mathfrak{r}}_{\mathfrak{1}}$ and ${\mathfrak{s}}_{\mathfrak{1}}$ explicit points of $\mathcal{W}$. $\mathfrak{h}$ is injective ⇒ $\mathfrak{h}$(${\mathfrak{s}}_{\mathfrak{1}}$) ≠ $\mathfrak{h}$(${\mathfrak{r}}_{\mathfrak{1}}$). As $\mathcal{Y}$ is ${\tau }_g$-$S_g^{*}$-$T_0$, there is a $\mathcal{F}\in {\tau }_g$-$S_g^{*}$O($\mathcal{Y}$) such that $\mathfrak{h}$(${\mathfrak{r}}_{\mathfrak{1}}$) ∈ $\mathcal{F}$, $\mathfrak{h}$(${\mathfrak{s}}_{\mathfrak{1}}$) ∉ $\mathcal{F}$, or there is a ${\mathcal{F}}_{\beta }\in {\tau }_g$-$S_g^{*}$O($\mathcal{Y}$) such that $\mathfrak{h}$(${\mathfrak{s}}_{\mathfrak{1}}$) ∈ ${\mathcal{F}}_{\beta }$, $\mathfrak{h}$(${\mathfrak{r}}_{\mathfrak{1}}$) ∉ ${\mathcal{F}}_{\beta }$ with $\mathfrak{h}$(${\mathfrak{s}}_{\mathfrak{1}}$) ≠ $\mathfrak{h}$(${\mathfrak{r}}_{\mathfrak{1}}$). As $\mathfrak{h}$ is ${\tau }_g$-$S_g^{*}$-irresolute, then ${\mathfrak{h}}^{-1}$($\mathcal{F}$) ∈ ${\tau }_g$-$S_g^{*}$O($\mathcal{W}$), and there is a ${\mathfrak{h}}^{-1}$(${\mathfrak{r}}_{\mathfrak{1}}$) ∈ $\mathcal{F}$, ${\mathfrak{h}}^{-1}$(${\mathfrak{s}}_{\mathfrak{1}}$) ∉ $\mathcal{F}$ or ${\mathfrak{h}}^{-1}$(${\mathcal{F}}_{\beta }$) ∈ ${\tau }_g$-$S_g^{*}$O($\mathcal{W}$) ⇒ ${\mathfrak{h}}^{-1}$(${\mathfrak{s}}_{\mathfrak{1}}$) ∈ ${\mathcal{F}}_{\beta }$, ${\mathfrak{h}}^{-1}$(${\mathfrak{r}}_{\mathfrak{1}}$) ∉ ${\mathcal{F}}_{\beta }$. Hence, $\mathcal{W}$ is a ${\tau }_g$-$S_g^{*}$-$T_0$ space. □

Theorem 16.

Assume $\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-$T_1$ iff ${\mathfrak{s}}_{\mathfrak{1}}\in \mathcal{W}$ singleton $\left\{{\mathfrak{s}}_{\mathfrak{1}}\right\}$ ∈ ${\tau }_g$-$S_g^{*}$C($\mathcal{W}$).

Proof. 

Assume $\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-$T_1$, ${\mathfrak{r}}_{\mathfrak{1}}\in \mathcal{W}$. Then ${\mathfrak{s}}_{\mathfrak{1}}\in \mathcal{W}$ - {${\mathfrak{r}}_{\mathfrak{1}}$} ⇒ ${\mathfrak{r}}_{\mathfrak{1}}\ne {\mathfrak{s}}_{\mathfrak{1}}\in \mathcal{W}$. However, $\mathcal{W}$ is a ${\tau }_g$-$S_g^{*}$-$T_1$ space ⇒ there is a $\mathcal{F}$, ${\mathcal{F}}_{\beta }\in {\tau }_g$-$S_g^{*}$O($\mathcal{W}$) ⇒ ${\mathfrak{r}}_{\mathfrak{1}}$ ∉ $\mathcal{F}$, ${\mathfrak{s}}_{\mathfrak{1}}\in {\mathcal{F}}_{\beta }$ ⊆ ($\mathcal{W}$ - {${\mathfrak{r}}_{\mathfrak{1}}$}). Furthermore, ${\mathfrak{s}}_{\mathfrak{1}}\in {\mathcal{F}}_{\beta }$ ⊆ ($\mathcal{W}$ - {${\mathfrak{r}}_{\mathfrak{1}}$}) ⇒ ($\mathcal{W}$ - {${\mathfrak{r}}_{\mathfrak{1}}$}) ∈ ${\tau }_g$-$S_g^{*}$O($\mathcal{W}$). Thus, {${\mathfrak{r}}_{\mathfrak{1}}$} is ${\tau }_g$-$S_g^{*}$-closed. Conversely, consider that ${\mathfrak{r}}_{\mathfrak{1}}\ne {\mathfrak{s}}_{\mathfrak{1}}\in \mathcal{W}$. Then {${\mathfrak{r}}_{\mathfrak{1}}$} and {${\mathfrak{s}}_{\mathfrak{1}}$} are ${\tau }_g$-$S_g^{*}$-closed sets, and {${\mathfrak{r}}_{\mathfrak{1}}$${\mathrm{\}}}^c$ is ${\tau }_g$-$S_g^{*}$-open. Certainly, {${\mathfrak{r}}_{\mathfrak{1}}$} ∉ {${\mathfrak{r}}_{\mathfrak{1}}$${\mathrm{\}}}^c$ and {${\mathfrak{s}}_{\mathfrak{1}}$} ∈ {${\mathfrak{r}}_{\mathfrak{1}}$${\mathrm{\}}}^c$. Similarly {${\mathfrak{s}}_{\mathfrak{1}}$${\mathrm{\}}}^c$ is ${\tau }_g$-$S_g^{*}$-open, {${\mathfrak{s}}_{\mathfrak{1}}$} ∉ {${\mathfrak{s}}_{\mathfrak{1}}$${\mathrm{\}}}^c$ and {${\mathfrak{r}}_{\mathfrak{1}}$} ∈ {${\mathfrak{s}}_{\mathfrak{1}}$${\mathrm{\}}}^c$. Thus $\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-$T_1$ space. □

Theorem 17.

Let $\mathfrak{h}$:$\mathcal{W}$ → $\mathcal{Y}$. Then the subsequent findings are valid:

(i) 

If $\mathfrak{h}$ is ${\tau }_g$-$S_g^{*}$-continuous and injective, $\mathcal{Y}$ is ${\tau }_g$-$T_1$ ⇒ $\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-$T_1$.

(ii) 

If $\mathfrak{h}$ is ${\tau }_g$-$S_g^{*}$-continuous and injective, $\mathcal{Y}$ is ${\tau }_g$-$T_2$ ⇒ $\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-$T_2$.

(iii) 

If $\mathfrak{h}$ is ${\tau }_g$-$S_g^{*}$-irresolute and injective, $\mathcal{Y}$ is ${\tau }_g$-$S_g^{*}$-$T_2$ ⇒ $\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-$T_2$.

Proof. 

(i) Assume ${\mathfrak{r}}_{\mathfrak{1}}\ne {\mathfrak{s}}_{\mathfrak{1}}$, where ${\mathfrak{r}}_{\mathfrak{1}}$, ${\mathfrak{s}}_{\mathfrak{1}}\in \mathcal{W}$. Then $\mathfrak{h}$(${\mathfrak{r}}_{\mathfrak{1}}$) = ${\mathfrak{r}}_{\mathfrak{2}}$ and $\mathfrak{h}$(${\mathfrak{s}}_{\mathfrak{1}}$) = ${\mathfrak{s}}_{\mathfrak{2}}$. Additionally, $\mathfrak{h}$(${\mathfrak{r}}_{\mathfrak{1}}$) ≠ $\mathfrak{h}$(${\mathfrak{s}}_{\mathfrak{1}}$). ($\mathcal{Y}$, ${\tau }_{g2}$) ${\tau }_g$-$T_1$ ⇒ ${\mathfrak{r}}_{\mathfrak{2}}\in {\mathcal{M}}_1$, ${\mathfrak{s}}_{\mathfrak{2}}$ ∉ ${\mathcal{M}}_1$ and ${\mathfrak{s}}_{\mathfrak{2}}\in {\mathcal{N}}_1$, ${\mathfrak{r}}_{\mathfrak{2}}$ ∉ ${\mathcal{N}}_1$. Then ${\mathfrak{r}}_{\mathfrak{1}}\in {\mathfrak{h}}^{-1}$(${\mathcal{M}}_1$), ${\mathfrak{r}}_{\mathfrak{1}}$ ∉ ${\mathfrak{h}}^{-1}$(${\mathcal{N}}_1$) and ${\mathfrak{s}}_{\mathfrak{1}}\in {\mathfrak{h}}^{-1}$(${\mathcal{N}}_1$), ${\mathfrak{s}}_{\mathfrak{1}}$ ∉ ${\mathfrak{h}}^{-1}$(${\mathcal{M}}_1$). According to the definition of ${\tau }_g$-$S_g^{*}$-continuity, ${\mathfrak{h}}^{-1}$(${\mathcal{M}}_1$) and ${\mathfrak{h}}^{-1}$(${\mathcal{N}}_1$) ∈ ${\tau }_g$-$S_g^{*}$O($\mathcal{W}$). For ${\mathfrak{r}}_{\mathfrak{1}}\ne {\mathfrak{s}}_{\mathfrak{1}}$, ${\mathfrak{r}}_{\mathfrak{1}}$, ${\mathfrak{s}}_{\mathfrak{1}}\in \mathcal{W}$ ⇒ ${\mathfrak{r}}_{\mathfrak{1}}\in {\mathfrak{h}}^{-1}$(${\mathcal{M}}_1$), ${\mathfrak{r}}_{\mathfrak{1}}$ ∉ ${\mathfrak{h}}^{-1}$(${\mathcal{N}}_1$) and ${\mathfrak{s}}_{\mathfrak{1}}\in {\mathfrak{h}}^{-1}$(${\mathcal{N}}_1$), ${\mathfrak{s}}_{\mathfrak{1}}$ ∉ ${\mathfrak{h}}^{-1}$(${\mathcal{M}}_1$). Thus, ($\mathcal{W}$, ${\tau }_{g1}$) is a ${\tau }_g$-$S_g^{*}$-$T_1$ space. In the same way, (ii) and (iii) can be proven. □

Theorem 18.

The subsequent statements are equivalent.

1. 

$\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-$T_2$.

2. 

If ${\mathfrak{r}}_{\mathfrak{1}}\in \mathcal{W}$, then ${\mathfrak{r}}_{\mathfrak{1}}\ne {\mathfrak{s}}_{\mathfrak{1}}$, there is a $\mathcal{U}$ containing ${\mathfrak{r}}_{\mathfrak{1}}$ and ${\mathfrak{s}}_{\mathfrak{1}}\notin {\tau }_g$-$S_g^{*}$Cl($\mathcal{U}$).

Proof. 

(1) ⇒ (2) Take ${\mathfrak{r}}_{\mathfrak{1}}\in \mathcal{W}$ and ${\mathfrak{s}}_{\mathfrak{1}}\in \mathcal{W}$ with ${\mathfrak{r}}_{\mathfrak{1}}\ne {\mathfrak{s}}_{\mathfrak{1}}$. There is a disjoint set $\mathcal{U}$ and $\mathcal{V}\in {\tau }_g$-$S_g^{*}$O($\mathcal{W}$) such that ${\mathfrak{r}}_{\mathfrak{1}}\in \mathcal{U}$ and ${\mathfrak{s}}_{\mathfrak{1}}\in \mathcal{V}$. Then ${\mathfrak{r}}_{\mathfrak{1}}\in \mathcal{U}\subseteq {\mathcal{V}}^c$ and ${\mathcal{V}}^c\in {\tau }_g$-$S_g^{*}$C($\mathcal{W}$) and ${\mathfrak{s}}_{\mathfrak{1}}\in {\mathcal{V}}^c$ ⇒ ${\mathfrak{s}}_{\mathfrak{1}}\notin {\tau }_g$-$S_g^{*}$Cl($\mathcal{U}$). (2) ⇒ (1) Consider ${\mathfrak{r}}_{\mathfrak{1}}\in \mathcal{W}$ and ${\mathfrak{s}}_{\mathfrak{1}}\in \mathcal{W}$ with ${\mathfrak{r}}_{\mathfrak{1}}\ne {\mathfrak{s}}_{\mathfrak{1}}$ ⇒ there is a ${\tau }_g$-$S_g^{*}$-open $\mathcal{U}$ containing ${\mathfrak{r}}_{\mathfrak{1}}$ such that ${\mathfrak{s}}_{\mathfrak{1}}\notin {\tau }_g$-$S_g^{*}$Cl($\mathcal{U}$) ⇒ ${\mathfrak{s}}_{\mathfrak{1}}\in$ ($\mathcal{W}$ - (${\tau }_g$-$S_g^{*}$Cl($\mathcal{U}$))). ($\mathcal{W}$ - (${\tau }_g$-$S_g^{*}$Cl($\mathcal{U}$))) ∈ ${\tau }_g$-$S_g^{*}$O($\mathcal{W}$) and ${\mathfrak{r}}_{\mathfrak{1}}$ ∉ ($\mathcal{W}$ - (${\tau }_g$-$S_g^{*}$Cl($\mathcal{U}$))). Additionally, $\mathcal{U}$ ∩ ($\mathcal{W}$ - (${\tau }_g$-$S_g^{*}$Cl($\mathcal{U}$))) = $\varphi$. Therefore, $\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-$T_2$. □

3.3.2. ${\tau }_g$-$S_g^{*}$-Regular Space

Definition 24.

If ∀$\mathcal{F}\in {\tau }_g$-$S_g^{*}$C($\mathcal{W}$) and $\mathfrak{r}\notin \mathcal{F}$, there are disjoint open sets $\mathcal{E}$ and $\mathcal{D}$ such that $\mathcal{F}\subseteq \mathcal{E}$, $\mathfrak{r}\in \mathcal{D}$.

The relationship between ${\tau }_g$-$S_g^{*}$-regularity and ${\tau }_g$-regularity is as follows.

Theorem 19.

Every ${\tau }_g$-$S_g^{*}$-regular space is ${\tau }_g$-regular.

Proof. 

Considering $\mathcal{W}$ to be a ${\tau }_g$-$S_g^{*}$-regular space. Take $\mathcal{F}\in {\tau }_g$-$S_g^{*}$C($\mathcal{W}$) and $\mathfrak{r}$ ∉ $\mathcal{F}$. As $\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-regular, there is a pair of disjoint open sets $\mathcal{E}$ and $\mathcal{D}$ such that $\mathcal{F}$ ⊆ $\mathcal{E}$, $\mathfrak{r}\in \mathcal{D}$. Hence, $\mathcal{W}$ is ${\tau }_g$-regular. □

Remark 4.

Every ${\tau }_g$-regular space is not a ${\tau }_g$-$S_g^{*}$-regular space.

Example 5.

Let $\mathcal{W}$ = $\{e_1,f_1,d_1\}$ and ${\tau }_g$ = $\{\varphi , \mathcal{W}$, $\left\{d_1\right\}$, $\{e_1,f_1\}\}$. Hence, ($\mathcal{W}$, ${\tau }_g$) is ${\tau }_g$-regular but not a ${\tau }_g$-$S_g^{*}$-regular space. For $\left\{f_1\right\}\in {\tau }_g$-$S_g^{*}$C($\mathcal{W}$) and $\mathfrak{r}\notin \left\{f_1\right\}$, there are no disjoint open sets $\mathcal{E}$ and $\mathcal{D}$ such that $\left\{f_1\right\}\subseteq \mathcal{E}$, $\mathfrak{r}\in \mathcal{D}$.

Theorem 20.

Every ${\tau }_g$-regular space with a ${\tau }_g$-$S_g^{*}$Tc space is ${\tau }_g$-$S_g^{*}$-regular.

Proof. 

Considering $\mathcal{W}$ to be ${\tau }_g$-regular, ${\tau }_g$-$S_g^{*}$Tc. Assume $\mathcal{F}\in {\tau }_g$-$S_g^{*}$C($\mathcal{W}$) and $\mathfrak{r}\in \mathcal{W}$ such that $\mathfrak{r}$ ∉ $\mathcal{F}$. As $\mathcal{W}$ is a ${\tau }_g$-$S_g^{*}$Tc space, $\mathcal{F}$ is closed, and $\mathfrak{r}$ ∉ $\mathcal{F}$. Since $\mathcal{W}$ is ${\tau }_g$-regular, there is a pair of disjoint open sets $\mathcal{E}$ and $\mathcal{D}$ such that $\mathcal{F}$ ⊆ $\mathcal{E}$, $\mathfrak{r}\in \mathcal{D}$. Hence, $\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-regular. □

Theorem 21.

If $\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-regular, then it is a ${\tau }_g$-regular space.

Proof. 

According to this fact, every closed set belongs to ${\tau }_g$-$S_g^{*}$C($\mathcal{W}$). □

Theorem 22.

The subsequent statements are equivalent.

1. 

$\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-regular.

2. 

$\forall \mathfrak{r}\in \mathcal{W}$ and each ${\tau }_g$-$S_g^{*}$-open nbhd $\mathcal{U}$, there is an open nbhd ${\mathcal{N}}_1$ of $\mathcal{W}$ such that Cl(${\mathcal{N}}_1$) $\subseteq \mathcal{U}$.

Proof. 

(1) ⇒ (2) Assume $\mathcal{U}$ is a ${\tau }_g$-$S_g^{*}$-neighbourhood of $\mathfrak{r}$. There is a $\mathcal{E}\in {\tau }_g$-$S_g^{*}$O($\mathcal{W}$) such that $\mathfrak{r}\in \mathcal{E}$ ⊆ $\mathcal{U}$. Now ${\mathcal{E}}^c\in {\tau }_g$-$S_g^{*}$C($\mathcal{W}$) and $\mathfrak{r}$ ∉ ${\mathcal{E}}^c$. From (1), there is a $\mathcal{R}$, $\mathcal{S}$ such that ${\mathcal{E}}^c$ ⊆ $\mathcal{R}$, $\mathfrak{r}\in \mathcal{S}$, $\mathcal{R}$ ∩ $\mathcal{S}$ = $\varphi$. Thus, $\mathcal{S}$ ⊆ ${{\mathcal{M}}_1}^c$. Now Cl($\mathcal{S}$) ⊆ Cl(${\mathcal{R}}^c$) = ${\mathcal{E}}^c$ and ${\mathcal{E}}^c$ ⊆ $\mathcal{R}$ ⇒ ${\mathcal{R}}^c$ ⊆ $\mathcal{E}$ ⊆ $\mathcal{U}$. Thus, Cl($\mathcal{S}$) ⊆ $\mathcal{U}$. (2) ⇒ (1) Consider a ${\tau }_g$-$S_g^{*}$-closed $\mathcal{F}$ in $\mathcal{W}$ and $\mathfrak{r}$ ∉ $\mathcal{F}$ or $\mathfrak{r}$ ∈ ($\mathcal{F}$)^c and $\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-open ⇒ ($\mathcal{F}$)^c is ${\tau }_g$-$S_g^{*}$-nbhd of $\mathfrak{r}$. By hypothesis, there is an open nbhd ${\mathcal{N}}_1$ such that $\mathfrak{r}\in {\mathcal{N}}_1$, Cl(${\mathcal{N}}_1$) ⊆ ($\mathcal{F}$)^c ⇒ $\mathcal{F}$ ⊆ {$\mathcal{W}$ - Cl(${\mathcal{N}}_1$)} and ${\mathcal{N}}_1$ ∩ {$\mathcal{W}$ - Cl(${\mathcal{N}}_1$)} = $\varphi$. Thus, $\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-regular. □

Theorem 23.

Assume $\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-regular iff for every $\mathcal{E}$-$S_g^{*}$C($\mathcal{W}$) and point $\mathfrak{p}\in$ ($\mathcal{W}$ - $\mathcal{E}$), $\mathfrak{r}\in \mathcal{U}$, $\mathcal{E}\subseteq {\mathcal{N}}_1$ and Cl(${\mathcal{N}}_1$) ∩ Cl($\mathcal{U}$) = ϕ, where $\mathcal{U}$ and ${\mathcal{N}}_1$ are open sets.

Proof. 

Given that $\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-regular. Assume $\mathcal{E}\in {\tau }_g$-$S_g^{*}$C($\mathcal{W}$) and $\mathfrak{r}\notin \mathcal{E}$. Then $\mathfrak{p}$${\mathcal{M}}_1$ and $\mathcal{E}\subseteq {\mathcal{N}}_1$ and ${\mathcal{M}}_1\cap {\mathcal{N}}_1$ = $\varphi$ where ${\mathcal{M}}_1$ and ${\mathcal{N}}_1$ are open sets ⇒ ${\mathcal{M}}_1$ ∩ Cl(${\mathcal{N}}_1$) = $\varphi$. As $\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-regular, $\mathfrak{p}\in \mathcal{R}$ and Cl(${\mathcal{N}}_1$) ⊆ $\mathcal{S}$, $\mathcal{R}$ ∩ ${\mathcal{N}}_1$ = $\varphi$, where $\mathcal{R}$ and $\mathcal{S}$ are open. Furthermore, Cl($\mathcal{R}$) ∩ $\mathcal{S}$ = $\varphi$. $\mathcal{U}$ = ${\mathcal{M}}_1$ ∩ $\mathcal{R}$ ⇒ $\mathfrak{p}\in \mathcal{U}$, $\mathcal{E}$ ⊆ ${\mathcal{N}}_1$ and Cl(${\mathcal{N}}_1$) ∩ Cl($\mathcal{U}$) = $\varphi$, where ${\mathcal{N}}_1$ and $\mathcal{U}$ are open in $\mathcal{W}$. On the other hand, consider ${\mathcal{N}}_1$ and $\mathcal{U}$ are open sets. $\mathfrak{p}\in \mathcal{U}$, $\mathcal{E}$ ⊆ ${\mathcal{N}}_1$ and Cl(${\mathcal{N}}_1$) ∩ Cl($\mathcal{U}$) = $\varphi$ ∀ $\mathcal{E}\in {\tau }_g$-$S_g^{*}$C($\mathcal{W}$) and $\mathfrak{p}$ ∈ ($\mathcal{W}$ - $\mathcal{E}$) ⇒ $\mathfrak{p}\in \mathcal{U}$, $\mathcal{E}$ ⊆ ${\mathcal{N}}_1$ and $\mathcal{U}$ ∩ ${\mathcal{N}}_1$ = $\varphi$. Thus, $\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-regular. □

Theorem 24.

A subspace $\mathcal{Y}$ of ${\tau }_g$-$S_g^{*}$-regular ($\mathcal{W}$, ${\tau }_g$) is ${\tau }_g$-$S_g^{*}$-regular.

Proof. 

Obvious. □

Theorem 25.

Consider that $\mathfrak{h}$ is bijective and ${\tau }_g$-$S_g^{*}$-irresolute and an open map from ${\tau }_g$-$S_g^{*}$-regular $\mathcal{W}$ into $\mathcal{Y}$. Then $\mathcal{Y}$ is ${\tau }_g$-$S_g^{*}$-regular.

Proof. 

Let ${\mathfrak{r}}_{\mathfrak{1}}\in \mathcal{Y}$ and $\mathcal{F}\in {\tau }_g$-$S_g^{*}$C($\mathcal{W}$) and ${\mathfrak{r}}_{\mathfrak{1}}$ ∉ $\mathcal{F}$. Additionally, $\mathfrak{h}$ is ${\tau }_g$-$S_g^{*}$-irresolute. Then ${\mathfrak{h}}^{-1}$($\mathcal{F}$) ∈ ${\tau }_g$-$S_g^{*}$C($\mathcal{W}$). Now assume ${\mathfrak{r}}_{\mathfrak{1}}$ = $\mathfrak{h}$($\mathfrak{r}$). Then ${\mathfrak{h}}^{-1}$(${\mathfrak{r}}_{\mathfrak{1}}$) = $\mathfrak{r}$ and $\mathfrak{r}$ ∉ ${\mathfrak{h}}^{-1}$($\mathcal{F}$). If $\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-regular, then there is a $\mathcal{R}$ and $\mathcal{S}$ such that $\mathfrak{r}\in \mathcal{R}$ and ${\mathfrak{h}}^{-1}$($\mathcal{F}$) ⊆ $\mathcal{S}$, $\mathcal{R}$ ∩ $\mathcal{S}$ = $\varphi$. $\mathfrak{h}$ is open and bijective ⇒ ${\mathfrak{r}}_{\mathfrak{1}}\in \mathfrak{h}$($\mathcal{R}$), $\mathcal{F}$ ⊆ $\mathfrak{h}$($\mathcal{S}$) and $\mathfrak{h}$($\mathcal{R}$ ∩ $\mathcal{S}$) = $\mathfrak{h}$($\varphi$) = $\varphi$ ⇒ $\mathcal{Y}$ is ${\tau }_g$-$S_g^{*}$-regular. □

3.3.3. ${\tau }_g$-$S_g^{*}$-Normal Space

Definition 25.

Assume $\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-normal if for each pair $\mathcal{E}$, $\mathcal{D}\in {\tau }_g$-$S_g^{*}$C($\mathcal{W}$), there are open sets $\mathcal{R}$ and $\mathcal{S}$ in $\mathcal{W}$ such that $\mathcal{D}\subseteq \mathcal{R}$ and $\mathcal{E}\subseteq \mathcal{S}$.

Theorem 26.

Every ${\tau }_g$-$S_g^{*}$-normal is ${\tau }_g$-normal.

Proof. 

As $\mathcal{W}$ is a ${\tau }_g$-$S_g^{*}$-normal. Assume disjoint sets $\mathcal{E}$ and $\mathcal{D}$ in $\mathcal{W}$. Therefore, $\mathcal{E}$, $\mathcal{D}\in {\tau }_g$-$S_g^{*}$C($\mathcal{W}$). $\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-normal ⇒ there is a pair $\mathcal{F}$, $\mathcal{H}$ such that $\mathcal{D}$ ⊆ $\mathcal{F}$, $\mathcal{E}$ ⊆ $\mathcal{H}$. Thus, $\mathcal{W}$ is ${\tau }_g$-normal. □

Remark 5.

The subsequent illustration demonstrates that the converse of the above theorem is invalid.

Example 6.

Consider $\mathcal{W}$ = $\{e_1,f_1,d_1\}$ and ${\tau }_g$ = $\{\varphi , \mathcal{W}$, $\left\{f_1\right\}$, $\left\{d_1\right\}$, $\{f_1,d_1\}$, $\{e_1,f_1\}\}$. Here ($\mathcal{W}$, ${\tau }_g$) is ${\tau }_g$-normal but not a ${\tau }_g$-$S_g^{*}$-normal space. For disjoint sets $\left\{e_1\right\}$, $\{f_1,d_1\}\in {\tau }_g$-$S_g^{*}$C($\mathcal{W}$), there are no open sets $\mathcal{R}$ and $\mathcal{S}$ in $\mathcal{W}$.

Theorem 27.

If $\mathcal{Y}$ is ${\tau }_g$-normal and a ${\tau }_g$-$S_g^{*}$Tc space, then $\mathcal{Y}$ is ${\tau }_g$-$S_g^{*}$-normal.

Proof. 

Assume $\mathcal{Y}$ is ${\tau }_g$-normal. Consider disjoint set $\mathcal{E}$, $\mathcal{D}\in {\tau }_g$-$S_g^{*}$C($\mathcal{Y}$). As ${\tau }_g$-$S_g^{*}$Tc space, $\mathcal{E}$ and $\mathcal{D}$ are closed. Since $\mathcal{Y}$ is ${\tau }_g$-normal, there are disjoint open sets $\mathcal{R}$ and $\mathcal{S}$ in $\mathcal{Y}$ such that $\mathcal{E}$ ⊆ $\mathcal{R}$ and $\mathcal{D}$ ⊆ $\mathcal{S}$. Thus, $\mathcal{Y}$ is ${\tau }_g$-$S_g^{*}$-normal. □

Theorem 28.

Every ${\tau }_g$-$S_g^{*}$-normal is ${\tau }_g$-g-normal.

Proof. 

Since $\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-normal. Disjoint set $\mathcal{E}$, $\mathcal{D}\in {\tau }_g$-$S_g^{*}$C($\mathcal{Y}$) ⇒ there is a disjoint $\mathcal{R}$, $\mathcal{S}$ such that $\mathcal{E}$ ⊆ $\mathcal{R}$ and $\mathcal{D}$ ⊆ $\mathcal{S}$. Thus, $\mathcal{W}$ is ${\tau }_g$-g-normal. □

Remark 6.

Every ${\tau }_g$-g-normal space is not ${\tau }_g$-$S_g^{*}$-normal.

Example 7.

Let $\mathcal{W}$ = $\{e_1,f_1,d_1\}$ and ${\tau }_g$ = $\{\varphi , \mathcal{W}$, $\left\{f_1\right\}$, $\left\{d_1\right\}$, $\{f_1,d_1\}$, $\{e_1,d_1\}\}$. Here ($\mathcal{W}$, ${\tau }_g$) is ${\tau }_g$-g-normal but not a ${\tau }_g$-$S_g^{*}$-normal space. For disjoint ${\tau }_g$-$S_g^{*}$-closed sets $\left\{e_1\right\}$, $\{f_1,d_1\}$, there are no open sets $\mathcal{R}$ and $\mathcal{S}$ in $\mathcal{W}$.

Theorem 29.

Every ${\tau }_g$-$S_g^{*}$-normal space is ${\tau }_g$-w-normal.

Proof. 

Similar to Theorem 28. □

Remark 7.

The converse of the above theorem is invalid, as demonstrated by subsequent illustration.

Example 8.

Let $\mathcal{W}$ = $\{e_1,f_1,d_1\}$ and ${\tau }_g$ = $\{\varphi , \mathcal{W}$, $\left\{f_1\right\}$, $\left\{d_1\right\}$, $\{f_1,d_1\}$, $\{e_1,d_1\}\}$. Here ($\mathcal{W}$, ${\tau }_g$) is ${\tau }_g$-w-normal but not a ${\tau }_g$-$S_g^{*}$-normal space. For disjoint ${\tau }_g$-$S_g^{*}$-closed sets $\left\{e_1\right\}$, $\{f_1,d_1\}$, there are no open sets $\mathcal{R}$ and $\mathcal{S}$ in $\mathcal{W}$.

Theorem 30.

If $\mathcal{Y}$ is a ${\tau }_g$-$S_g^{*}$-closed subspace of ${\tau }_g$-$S_g^{*}$-normal $\mathcal{W}$, then $\mathcal{Y}$ is ${\tau }_g$-$S_g^{*}$-normal.

Proof. 

Assume $\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-normal and $\mathcal{Y}$ is a ${\tau }_g$-$S_g^{*}$-closed subspace. A pair of disjoint sets $\mathcal{E}$ and $\mathcal{D}\in {\tau }_g$-$S_g^{*}$C($\mathcal{Y}$) ⇒ there is a $\mathcal{F}$, $\mathcal{H}\in \mathcal{W}$ such that $\mathcal{E}$ ⊆ $\mathcal{F}$ and $\mathcal{D}$ ⊆ $\mathcal{H}$ ⇒ $\mathcal{F}$ ∩ $\mathcal{Y}$ and $\mathcal{H}$ ∩ $\mathcal{Y}$ are open in $\mathcal{Y}$. Furthermore, $\mathcal{E}$ ⊆ $\mathcal{F}$ and $\mathcal{D}$ ⊆ $\mathcal{H}$ ⇒ $\mathcal{E}$ ∩ $\mathcal{Y}$ ⊆ $\mathcal{Y}$ ∩ $\mathcal{F}$ and $\mathcal{Y}$ ∩ $\mathcal{D}$ ⊆ $\mathcal{Y}$ ∩ $\mathcal{H}$ and ($\mathcal{F}$ ∩ $\mathcal{Y}$) ∩ ($\mathcal{Y}$ ∩ $\mathcal{H}$) = $\mathcal{Y}$ ∩ ($\mathcal{F}$ ∩ $\mathcal{H}$) = $\varphi$. Thus, $\mathcal{Y}$ is ${\tau }_g$-$S_g^{*}$-normal. □

Theorem 31.

The subsequent statements in ($\mathcal{W}$, ${\tau }_g$) are equivalent.

(1) 

$\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-normal.

(2) 

$\mathcal{E}\subseteq {\mathcal{U}}_1\subseteq$ Cl(${\mathcal{U}}_1$) $\subseteq {\mathcal{U}}_2$ for each $\mathcal{E}\in {\tau }_g$-$S_g^{*}$C($\mathcal{W}$) and each ${\mathcal{U}}_2\in {\tau }_g$-$S_g^{*}$O($\mathcal{W}$) with $\mathcal{E}\subseteq {\mathcal{U}}_2$, where ${\mathcal{U}}_1$ is an open set.

(3) 

For any disjoint set $\mathcal{E}$, $\mathcal{D}\in {\tau }_g$-$S_g^{*}$C($\mathcal{W}$), there is an open set ${\mathcal{U}}_1$ such that $\mathcal{E}\subseteq {\mathcal{U}}_1$ and Cl(${\mathcal{U}}_1$) $\mathrm{\cap } \mathcal{D}$ = ϕ.

(4) 

For each disjoint set $\mathcal{E}$, $\mathcal{D}\in {\tau }_g$-$S_g^{*}$C($\mathcal{W}$), there are open sets ${\mathcal{U}}_1$, ${\mathcal{U}}_2$ such that $\mathcal{E}\subseteq {\mathcal{U}}_2$, $\mathcal{D}\subseteq {\mathcal{U}}_1$ and Cl(${\mathcal{U}}_2$) ∩ Cl(${\mathcal{U}}_1$) = ϕ.

Proof. 

(1) ⇒ (2) Take $\mathcal{E}\in {\tau }_g$-$S_g^{*}$C($\mathcal{W}$). ${\mathcal{U}}_2\in {\tau }_g$-$S_g^{*}$C($\mathcal{W}$) such that $\mathcal{E}\in {\mathcal{U}}_2$ ⇒ $\mathcal{E}$ and ($\mathcal{W}$ - ${\mathcal{U}}_2$) are two disjoint sets of ${\tau }_g$-$S_g^{*}$C($\mathcal{W}$). According to the ${\tau }_g$-$S_g^{*}$-normal definition, $\mathcal{E}\subseteq {\mathcal{U}}_1$ and ($\mathcal{W}$ - ${\mathcal{U}}_2$) ⊆ ${\mathcal{U}}_3$, where ${\mathcal{U}}_1$ and ${\mathcal{U}}_3$ are open disjoint sets. Therefore, ${\mathcal{U}}_1$ ⊆ ($\mathcal{W}$ - ${\mathcal{U}}_3$) and ${\mathcal{U}}_1$ ∩ ${\mathcal{U}}_3$ = $\varphi$ ⇒ ${\mathcal{U}}_1$ ⊆ ($\mathcal{W}$ - ${\mathcal{U}}_3$) and Cl(${\mathcal{U}}_1$) ⊆ ($\mathcal{W}$ - ${\mathcal{U}}_3$) ⊆ ${\mathcal{U}}_2$ ⇒ Cl(${\mathcal{U}}_1$) ⊆ ${\mathcal{U}}_2$. Thus, $\mathcal{E}$ ⊆ ${\mathcal{U}}_1$ ⊆ Cl(${\mathcal{U}}_1$) ⊆ ${\mathcal{U}}_2$. (2) ⇒ (3) Consider disjoint sets $\mathcal{E}$, $\mathcal{D}\in {\tau }_g$-$S_g^{*}$C($\mathcal{W}$), $\mathcal{E}$ ⊆ ($\mathcal{W}$ - $\mathcal{D}$) where $\mathcal{E}$ and ($\mathcal{W}$ - $\mathcal{D}$) are ${\tau }_g$-$S_g^{*}$-closed and ${\tau }_g$-$S_g^{*}$-open sets in $\mathcal{W}$. From (2), there is an open set ${\mathcal{U}}_1$ and $\mathcal{E}$ ⊆ ${\mathcal{U}}_1$ ⊆ Cl(${\mathcal{U}}_1$) ⊆ ($\mathcal{W}$ - $\mathcal{D}$). Cl(${\mathcal{U}}_1$) ⊆ ($\mathcal{W}$ - $\mathcal{D}$) ⇒ $\mathcal{D}$) ∩ Cl(${\mathcal{U}}_1$) = $\varphi$. Thus, $\mathcal{E}$ ⊆ ${\mathcal{U}}_1$ and Cl(${\mathcal{U}}_1$) ⊆ $\mathcal{D}$ = $\varphi$. (3) ⇒ (4) Consider disjoint sets $\mathcal{E}$, $\mathcal{D}\in {\tau }_g$-$S_g^{*}$C($\mathcal{W}$). By (3), there is an open set ${\mathcal{U}}_2$ such that $\mathcal{E}$ ⊆ ${\mathcal{U}}_2$ and Cl(${\mathcal{U}}_2$) ∩ $\mathcal{D}$ = $\varphi$. However, Cl(${\mathcal{U}}_2$) is closed ⇒ a ${\tau }_g$-$S_g^{*}$-closed set in $\mathcal{W}$. As Cl(${\mathcal{U}}_2$), $\mathcal{D}\in {\tau }_g$-$S_g^{*}$C($\mathcal{W}$). Then, by (3), $\mathcal{D}$ ⊆ ${\mathcal{U}}_1$ and Cl(${\mathcal{U}}_2$) ∩ Cl(${\mathcal{U}}_1$) = $\varphi$, where ${\mathcal{U}}_1$ is an open set in $\mathcal{W}$. (4) ⇒ (1) Assume disjoint sets $\mathcal{E}$, $\mathcal{D}\in {\tau }_g$-$S_g^{*}$C($\mathcal{W}$). Then by hypothesis there are open sets ${\mathcal{U}}_1$ and ${\mathcal{U}}_2$ such that $\mathcal{E}$ ⊆ ${\mathcal{U}}_2$ and $\mathcal{D}$ ⊆ ${\mathcal{U}}_1$, Cl(${\mathcal{U}}_2$) ∩ Cl(${\mathcal{U}}_1$) = $\varphi$ ⇒ $\mathcal{E}$ ⊆ ${\mathcal{U}}_2$ and $\mathcal{D}$ ⊆ ${\mathcal{U}}_1$. Thus, $\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-normal. □

Theorem 32.

Assume a mapping $\mathfrak{h}$:$\mathcal{W}$ → $\mathcal{Y}$. If $\mathfrak{h}$ is bijective, open, and ${\tau }_g$-$S_g^{*}$-irresolute from a ${\tau }_g$-$S_g^{*}$-normal $\mathcal{W}$ onto $\mathcal{Y}$, then $\mathcal{Y}$ is ${\tau }_g$-$S_g^{*}$-normal.

Proof. 

Assume disjoint sets $\mathcal{E}$, $\mathcal{D}\in {\tau }_g$-$S_g^{*}$C($\mathcal{W}$). Since $\mathfrak{h}$ is ${\tau }_g$-$S_g^{*}$-irresolute, ${\mathfrak{h}}^{-1}$($\mathcal{E}$) and ${\mathfrak{h}}^{-1}$($\mathcal{D}$) are in ${\tau }_g$-$S_g^{*}$C($\mathcal{W}$). $\mathcal{W}$ is ${\tau }_g$-$S_g^{*}$-normal ⇒ ${\mathfrak{h}}^{-1}$($\mathcal{E}$) ⊆ ${\mathcal{U}}_2$ and ${\mathfrak{h}}^{-1}$($\mathcal{D}$) ⊆ ${\mathcal{U}}_1$, where ${\mathcal{U}}_1$ and ${\mathcal{U}}_2$ are open in $\mathcal{W}$. Also, as $\mathfrak{h}$ is bijective and open, $\mathfrak{h}$(${\mathcal{U}}_2$) and $\mathfrak{h}$(${\mathcal{U}}_1$) are open and $\mathcal{E}$ ⊆ $\mathfrak{h}$(${\mathcal{U}}_2$), $\mathcal{D}$ ⊆ $\mathfrak{h}$(${\mathcal{U}}_1$). Thus, $\mathcal{Y}$ is ${\tau }_g$-$S_g^{*}$-normal. □

Theorem 33.

The subsequent statements in ($\mathcal{W}$, ${\tau }_g$) are equivalent.

(1) 

$\mathcal{W}$ is ${\tau }_g$-g-normal.

(2) 

There are disjoint open sets ${\mathcal{U}}_1$, ${\mathcal{U}}_2\in {\tau }_g$-$S_g^{*}$O($\mathcal{W}$) such that $\mathcal{E}\subseteq {\mathcal{U}}_2$, $\mathcal{D}\subseteq {\mathcal{U}}_1$ for each disjoint set $\mathcal{E}$ and $\mathcal{D}$.

Proof. 

(1) ⇒ (2) Let $\mathcal{W}$ is ${\tau }_g$-g-normal. Assume disjoint sets $\mathcal{E}$, $\mathcal{D}$ of $\mathcal{W}$. Since ($\mathcal{W}$, ${\tau }_g$) is ${\tau }_g$-g-normal, there are disjoint ${\tau }_g$-g-open sets ${\mathcal{U}}_1$ and ${\mathcal{U}}_2$ such that $\mathcal{E}$ ⊆ ${\mathcal{U}}_2$, $\mathcal{D}$ ⊆ ${\mathcal{U}}_1$ ⇒ ${\mathcal{U}}_1$, ${\mathcal{U}}_2\in {\tau }_g$-$S_g^{*}$O($\mathcal{W}$) with $\mathcal{E}$ ⊆ ${\mathcal{U}}_2$, $\mathcal{D}$ ⊆ ${\mathcal{U}}_1$ and ${\mathcal{U}}_1$ ∩ ${\mathcal{U}}_2$ = $\varphi$. (2) ⇒ (1) Assume two disjoint ${\tau }_g$-$S_g^{*}$-closed sets $\mathcal{E}$, $\mathcal{D}\in {\tau }_g$-$S_g^{*}$C($\mathcal{W}$). By hypothesis, $\mathcal{E}$ ⊆ ${\mathcal{U}}_2$, $\mathcal{D}$ ⊆ ${\mathcal{U}}_1$ and ${\mathcal{U}}_1$ ∩ ${\mathcal{U}}_2$ = $\varphi$, where ${\mathcal{U}}_1$ and ${\mathcal{U}}_2$ are disjoint sets in ${\tau }_g$-$S_g^{*}$O($\mathcal{W}$). Since $\mathcal{E}$ ⊆ ${\tau }_g$-gInt(${\mathcal{U}}_2$), $\mathcal{D}$ ⊆ ${\tau }_g$-gInt(${\mathcal{U}}_1$) and ${\tau }_g$-gInt(${\mathcal{U}}_2$) ∩ ${\tau }_g$-gInt(${\mathcal{U}}_1$) = $\varphi$. Thus, ($\mathcal{W}$, ${\tau }_g$) is ${\tau }_g$-g-normal. □

4. Methodology

A theoretical method is used in this study to examine the separation axioms, connectedness, and compactness in generalized topological spaces. The following is the structure of the methodology:

○

Literature Review: Existing research on the separation axioms, connectedness, and compactness in generalized topological spaces is thoroughly reviewed. The fundamental framework for applying these ideas to the $S_g^{*}$-compact and $S_g^{*}$-connected spaces in more comprehensive frameworks is laid forth in this review.

○

Definition and Evolution: In the generalized topological contexts, new definitions and characteristics of connectedness and compactness are put forward. To make sure that these definitions align with the fundamental structure of generalized topologies, they are thoroughly examined.

○

Analysis of the Separation Axioms: Within the same generalized frames, the separation axioms $T_0$, $T_1$, and $T_2$ are examined.

○

Theoretical Proofs: The specified properties and definitions are supported by formal proofs, demonstrating their applicability and validity. The associations between compactness, connectedness, and separation properties are analyzed.

○

Comparative Analysis: To demonstrate the differences, analogies, and advantages of the generalized approach, a comparison between these expanded notions and their classical equivalents is analyzed.

○

Conclusions and Consequences: The outcome findings are synthesized to provide a comprehensive understanding of the behavior of $S_g^{*}$-compact space, $S_g^{*}$-connected space, and separation axioms within generalized topological contexts. The implications of these results for future research in generalized topology are discussed.

5. Conclusions

This study provides a comprehensive exploration of compactness, connectedness, and separation axioms within the frameworks of generalized topological spaces. By extending classical notions of compactness and connectedness to $S_g^{*}$-compact and $S_g^{*}$-connected spaces, the research highlights the versatility and depth of these properties in generalized settings. The definitions and properties introduced are validated through rigorous theoretical analysis, offering new insights into the structural characteristics of these spaces. In addition, the investigation of separation axioms $S_g^{*}$-$T_0$, $S_g^{*}$-$T_1$, and $S_g^{*}$-$T_2$ within generalized topology underscores their critical role in classifying spaces based on point–set distinguishability. Although no direct relationships between $S_g^{*}$-compactness, $S_g^{*}$-connectedness, and separation axioms were established, their independent study provides a clearer understanding of the topological landscape in generalized frameworks. This work contributes to the growing body of research in generalized topology, laying the groundwork for future studies that may further refine these concepts or explore their applications in advanced mathematical theories and related disciplines. It emphasizes the importance of extending classical topological ideas to accommodate more complex and generalized structures.