Some Variants of Normal ˇCech Closure Spaces via Canonically Closed Sets

: New generalizations of normality in ˇCech closure space such as π -normal, weakly π normal and κ -normal are introduced and studied using canonically closed sets. It is observed that the class of κ -normal spaces contains both the classes of weakly π -normal and almost normal ˇCech closure spaces.


Introduction and Preliminaries
It is evident from the literature that topological structures which are more general than the classical topology are more suitable for the study of digital topology, image processing, network theory, pattern recognition and related areas. Various generalized structures such as closure spaces, generalized closure spaces,Čech closure spaces, generalized topologies (GT), weak structures (WS), Generalized neighborhood systems (GNS) etc. were introduced and studied in the past (see [1][2][3][4][5]). However, recentlyČech closure spaces attracted the attention of researchers due to its possibility of application in other applied fields discussed above. Usefulness of thisČech closure setting in variety of allied fields such as digital topology, computer graphics, image processing and pattern recognition are available in the literature [6][7][8][9].Čech closure space was defined byČech [1], are obtained from Kuratowski [10] closure operator by omitting the idempotent condition. In this setting Galton [11] studied the motion of an object in terms of a function giving its position at each time and systematically investigated what a continuous motion looks like. J. Šlapal [6] observed that this structure is more suitable than others for application in digital topology becauseČech closure spaces are well-behaved with respect to connectedness. Allam et al. [12,13] introduced a new method for generating closure spaces via a binary relation which was subsequently used by G. Liu [14] to establish a one-to-one correspondence between quasi discrete closures and reflexive relation. Furthermore, J. Šlapal and John L. Pfaltz [15] studied network structures via associated closure operators. Higher separation axioms inČech closure space was introduced by Barbel M. R. Stadler and F. Peter Stadler [16] in 2003 and discussed the concept of Urysohn functions, normal, regular, completely normal etc. in the form of neighborhood. In 2018 Gupta and Das [17] introduced higher separation axioms via relation. Since normality is an important topological property, many weak variants of normality introduced and studied in the past to properly study normality in general topology (See [18][19][20][21][22]). In the present paper, we introduced some variants of normality inČech closure space as π-normal, weakly π-normal and κ-normal using canonically closed sets. It is observed that some characterizations of normality and almost normality which holds in topological spaces may not hold in Cech closure spaces. Further relation between newly defined notions and already defined notions was also investigated.
A closure space is a pair (X, cl), where X is any set and closure cl : P(X) → P(X) is a function associating with each subset A ⊆ X to a subset cl(A) ⊆ X, called the closure of A, such that cl(∅) = ∅, A ⊆ cl(A), cl(A ∪ B) = cl(A) ∪ cl(B). With any closure cl for a set X there is associated the interior operation int cl , usually denoted by int, which is a single-valued relation on P(X) ranging in P(X) such that for each A ⊆ X, int cl (A) = X − cl(X − A). The set int cl (A) is called the interior of A in (X, cl). Remark 1. The notion of normality defined above in the Definition 1 is different from the notion of normality defined in [1]. A closure space is said to be normal [1] if every pair of sets with disjoint closures are separated by disjoint neighborhoods. The disjoint sets considered byČech for separation in the definition of normality are not necessarily closed sets and neighborhoods need not be open. Throughout the present paper, we have taken the notion of normality only in the sense of Definition 1.

Lemma 1.
[1] If U and V are subsets of a closure space (X, cl) such that U ⊆ V then cl(U) ⊆ cl(V). Theorem 1. [23] Suppose (X, cl) is aČech closure space such that int(cl(U)) is canonically open for every open set U. Then (X, cl) is weakly normal and almost normal implies (X, cl) is normal.

Variants of NormalČech Closure Space
Definition 2. Let (X, cl) be aČech closure space then A is said to be π-closed if it is equal to the intersection of two canonically closed set.
, c, d} be the set and define cl : Here, the set A = {a} is π-closed as it is the intersection of two canonically closed set i.e., {a, c, d} and {a, b} but {a} is not canonically closed.
In thisČech closure space, cl(A) = {d} = A is closed but not π-closed as it is not equal to the intersection of two canonically closed set.
The implications in Figure 1 are obvious from the definitions. However, none of these implications is reversible as shown in the above example.

Definition 3.
AČech closure space (X, cl) is π-normal if for every two disjoint closed sets one of which is π-closed there exist two disjoint open sets U and V containing the closed set and the π-closed set respectively.
It is obvious that in aČech closure space (X, cl), every normal space is π-normal. However, the converse need not be true as shown below.

Example 2.
AČech closure space which is π-normal but not normal. Let X = Y ∪ {p, q} be an infinite set, then any set A ∈ P(X) is one of the following four types of sets: is finite and A contains either p or q. Type-IV: (Y − A) is finite and A contains both p and q.
Define cl : P(X) → P(X) by In thisČech closure space, type-I and type-IV sets are closed sets. A set U is open if U is an infinite set containing p and/or q whose complement is finite. Additionally, a finite set U in Y whose complement is infinite is an open set in X. In this space only two types of sets are canonically closed. i.e., (1) Every finite set in Y is canonically closed (2) a set containing both p and q whose complement is finite in Y is canonically closed. This space is π-normal but not normal because for two disjoint closed sets  Following examples establish that the notion of weak normality defined earlier, and the notion of π-normality are independent notions.

Example 4.
A space which is weakly normal but not π-normal. Let X be the set of positive integers. Define cl : P(X) → P(X) as defined in Example 3. Here, thě Cech closure space (X, cl) is weakly normal but not π-normal as shown in Example 3.   T 2 if any two distinct points x and y are separated.

Remark 2.
In aČech closure space, every normal T 1 space is regular and T 2 . but if we replace normal by π-normal then the result need not be true. Consider the space defined in Example 2 which is π-normal and T 1 but neither T 2 nor regular. The space is not T 2 because disjoint points 'p' and 'q' cannot be separated and is not regular because for closed set A = C ∪ {p} where C is finite in Y and a point 'q' there does not exist disjoint open sets satisfying the required condition.

Definition 6. [24]
AČech closure space is said to be almost regular if for canonically closed set cl(int(A)) = A and a point x / ∈ cl(int(A)) there exist disjoint open sets U and V such that x ∈ U and cl(int(A)) ⊆ V.

Theorem 4.
In aČech closure space, every π-normal T 1 space is almost regular.
Proof. let cl(int(A)) = A be a canonically closed set and x / ∈ cl(int(A)) be a point. Since the space is a T 1Č ech closure space, the singleton set {x} is closed. As every canonically closed set is π-closed, by π-normality there exist disjoint open sets U and V such that cl(int(A) ⊆ U and {x} ⊆ V. Hence (X, cl) is an almost regularČech closure space.

Definition 7.
AČech closure space is said to be weakly π-normal if for two disjoint π-closed sets there exist disjoint open sets separating them.

Definition 8.
AČech closure space is said to be κ-normal if for two disjoint canonically closed sets A and B there exist disjoint open sets U and V containing A and B respectively.
From the definitions it is observed that every π-normal space is weakly π-normal, every weakly π-normal space as well as every almost normal space is κ-normal. Thus, the implications in Figure 2 are obvious but none of them is reversible which is exhibited below by Examples.
almost normal weakly π-normal / / κ-normal  Here, (X, cl) is aČech closure space which is T 1 almost normal but not regular because for closed set cl(A) = A and a point disjoint from the closed set A there does not exist disjoint open sets separating them.
The following theorem directly follows from the Theorem 1. full subcategory. Perfilieva et al. [29] investigated the relationship between L-Fuzzy Cech closure spaces and L-Fuzzy co-topological spaces from the categorical viewpoint. Relational variants of categories related to L-Fuzzy closure spaces was studied in [30].
In this paper, we have defined and investigated few variants of normality inČech closure spaces using canonically closed sets. Normality is an important topological property, and its importance is due to its behaviour as it behaves differently from other separation axioms for subspaces and products. Additionally, the class of normal spaces are more general than the important class of compact Hausdorff spaces. Normality involves separation of closed sets by open sets. On the other hand, in digital image processing a picture needs to be segmented into subsets where relationship of these subset from other neighboring subsets and adjoining points plays a prominent role for the processing of images. Such types of relationships between sets/points are either geometrical or topological. Geometrical relation involves position of points whereas topological relation involves concepts such as adjacency, neighborhood, separation, connectedness and compactness. So, the possibility of application of the notions defined in this paper in digital topology and digital image processing cannot be ruled out.