Limit points and separation axioms with respect to supra semi-open sets

Sometimes we need to minimize the conditions of topology for different reasons such as obtaining more convenient structures to describe some real-life problems, or constructing some counterexamples whom show the interrelations between certain topological concepts, or preserving some properties under fewer conditions of those on topology. To contribute this research area, in this paper, we establish some new concepts on supra topological spaces using supra semi-open sets and give some characterizations of them. First, we introduce a concept of supra semi limit points of a set and study main properties, in particular, on the spaces that possess the difference property. Second, we define and investigate new separation axioms, namely supra semi Ti-spaces (i = 0, 1, 2, 3, 4) and give complete descriptions for each one of them. We provide some examples to show the relationships between them as well as with STi-space. 2020 Mathematics Subject Classifications: 54A05, 54C10, 54D10, 54D15


Introduction and preliminaries
A structure on a nonempty set X is a subset of its power set P (X), topological spaces is an example of structure satisfying three conditions. In fact, topological spaces have been generalized in many ways. Alexendroff [12], in 1940, developed abstract spaces where he strengthened the intersection condition. Mashhour et al. [21], in 1983, considered supra topological spaces by neglecting the intersection condition. Maki et al. [20], in 1996, presented a minimal structure as a collection contains the empty and universal sets. Császár [13], in 2002, introduced generalized topological spaces if it contains the empty set and is closed under a nonempty union; and in 2011, he [14] studied a weak structure as a collection contains the empty set. Al-Odhari [1], in 2015, defined infra topological spaces by dropping only the union condition.
Many aspects of such spaces have already been studied. Supra topological spaces is one of the most important developments of general topology in recent years. A family µ of subsets of a nonempty set X is called a supra topology if it satisfies two conditions: The first one is the empty and universal sets belong to µ; and the second one is it is closed under arbitrary union. Some authors remove the empty set from the first condition of a supra topology because it is obtained from the second condition as the union of an empty collection of sets. Mashhour et al. [21] studied some basic operators, continuity and separation axioms on supra topological spaces. Al-shami [2] investigated the classical topological notions such as limit points of a set, compactness, and separation axioms on the supra topological spaces. He [7] also studied paracompactness on supra topological spaces. It should be noted that the supra topological frame can be more convenient to solve some practical problems and to model some phenomena as pointed out in [19].
Some results via topology do not still valid via supra topology such as the distribution of the closure operator between the union of two sets and the distribution of the interior operator between the intersection of two sets. Also, the property of a compact subset of a T 2 -space is closed is invalid on the supra topologies.
To extend a class of supra open sets, the notion of supra α-open [15], supra pre-open [24], supra b-open [26], supra β-open [18], supra R-open [16] and supra semi-open sets [3] have been introduced and their main properties have been discussed. These generalizations of supra open sets were defined in a similar way of defining them on general topology. In other words, their definitions were formulated using supra interior and supra closure operators instead of interior and closure operators. These generalizations have been utilized to define new versions of compactness, connectedness and separation axioms, see, for example [5,6,17,22,25]. The class of supra R-open sets has been studied in [4,11] under the name of somewhere dense sets. Mustafa and Qoqazeh [23] took advantage of supra D-sets to define separation axioms on supra topological spaces. Recently, Al-shami and El-Shafei [9,10] have studied separation axioms on supra soft topological spaces and supra soft topological ordered spaces.
The layout of the paper is as following: In Section (2), we explore a concept of supra semi limit points of a set. In Section (3), we initiate new types of separation axioms using supra semi-open sets and illustrate the relationships between them with the help of examples. Section (4) concludes the paper with summary and further works.
In the rest of this section, we mention some definitions and results of supra topology and supra semi-open sets that make this study self-contained and easy to read. Definition 1. [21] A family µ of subsets of a nonempty set X is called a supra topology provided that the following two conditions hold.
(ii) µ is closed under arbitrary union.
Then the pair (X, µ) is called a supra topological space. Every element of µ is called a supra open set and its complement is called a supra closed set. Remark 1. (i) µ is called an associated supra topology with a topology τ if τ ⊆ µ.

Definition 2. [21] Let
A be a subset of (X, µ). Then int µ (A) is the union of all supra open sets contained in A and cl µ (A) is the intersection of all supra closed sets containing A.
If there is no confusion, we write int(A) and cl(A) in the places of int µ (A) and cl µ (A), respectively. If there is no confusion, we write sint(A) and scl(A) in the places of sint µ (A) and scl µ (A), respectively. (ii) supra semi-open (resp. supra semi-closed) if the image of each open (resp. closed) subset of X is a supra semi-open (resp. supra semi-closed) subset of Y .

Definition 6.
[2] Let A be a subset of (X, µ). The family µ A = {A G : G ∈ µ} is called a supra relative topology on A. A pair (A, µ A ) is called a supra subspace of (X, µ).

Definition 7.
[8] β is called a basis for a supra topology (X, µ) if every member of µ can be expressed as a union of elements of β.
.., n} be the collection of supra topological spaces.
is called a finite product supra spaces. Proposition 1. [8] Let A and B be two subsets of (X, µ) and (Y, ν), respectively. Then:

Limit points of a set with respect to supra semi-open sets
In this section, we introduce and study supra semi limit points of a set. We explore many properties of them and discuss their behaviour on the spaces that possess the difference property. For more illustration of the presented findings, some interesting examples are given.

Definition 9.
A subset A of (X, µ) is said to be a supra semi neighbourhood of x ∈ X provided that there is a supra semi open set F containing x such that x ∈ F ⊆ A.

Definition 10.
A point x ∈ X is said to be a supra semi limit point of a subset A of (X, µ) provided that every supra semi neighborhood of x contains at least one point of A other than x itself.
All supra semi limit points of A is said to be a supra semi derived set of A and is denoted by A s .
Proof. Straightforward. Corollary 1. We have the following results for every A, B ⊆ (X, µ).
The following example illustrates that the converse of the above proposition and corollary fails.    Theorem 1. Let A be a subset of (X, µ). Then the following results hold.
From (1) and (2), Since A A s is a supra semi-closed set containing A and scl(A) is the smallest supra semi-closed set containing A, then scl(A) ⊆ A A s . Therefore scl(A) = A A s .

Corollary 2. If
A is a supra semi-closed subset of (X, µ), then A s , (A s ) s , ((A s ) s ) s , ... are supra semi-closed sets.
Definition 11. A map g : (X, µ) → (Y, ν) is said to be: is a supra semi-open (resp. supra semi-closed) set in Y for every supra semi-open (resp. supra semi-closed) set in X.
(iii) supra semi -homeomorphism if it is bijective, supra semi -continuous and supra semi -open.
Proof. Let a ∈ (g(A)) s . Then there is a supra semi-open set H containing a such that Since g is bijective, then a ∈ g(A s ). Therefore g(A s ) ⊆ (g(A)) s . By reversing the preceding steps, we find that (g(A)) s ⊆ g(A s ). Hence, the proof is complete.
Definition 12. For a nonempty set X, a sub collection Λ of 2 X is said to have the difference property provided that G ∈ Λ implies that G \ {x} ∈ Λ.
The following two examples illustrate the existence and uniqueness of the difference property.
Then (N , µ) has the difference property. Also, it can be seen that the collection of supra semi-open subsets of (N , µ) coincides with the collection of supra open sets. Hence, (N , µ) has the difference property for the collection of supra semi-open sets.
Therefore (X, µ) does not have the difference property.
Theorem 3. If (X, µ) has the difference property for the collection of supra semi-open sets, then the following properties hold for A ⊆ X.
Since (X, µ) has the difference property for the collection of supra semi-open sets, then G \ {x} is a supra semi-open set. Therefore From (3) and (4), the desired result is proved.
(iii) Let A be a finite subset of X. Suppose that there exists an element x ∈ X such that x ∈ A s . Then for every supra semi-open set G containing x, we have G\{x} A = ∅. Therefore for every y ∈ A such that y = This implies that x ∈ A s . But this is a contradiction. Hence, it must be that A s = ∅.
We explain that the three properties mentioned in the above theorem need not be true if (X, µ) does not have the difference property for the collection of supra semi-open sets. Let A = {1, 3} be a subset of supra topological space given in Example (3). Then A s = N \ {1, 3}, (A s ) s = N and scl(A s ) = N . This leads to the following three properties.

Separation axioms with respect to supra semi-open sets
In this section, we utilize supra semi-open sets to introduce the concepts of supra semi regular, supra semi normal and SsT i -spaces (i = 0, 1, 2, 3, 4). We give some characterizations for each one of them and elucidate the relationships among themselves as well as with ST i -space. (iv) supra semi regular if for every supra semi-closed set F and each a ∈ F , there exist disjoint supra semi-open sets U and V containing F and a, respectively.
(v) supra semi normal if for every disjoint supra semi-closed sets F and H, there exist disjoint supra semi-open sets U and V containing F and H, respectively.
(vi) SsT 3 (resp. SsT 4 ) if it is both supra semi regular (resp. supra semi normal) and SsT 1 . (iii) For each a ∈ X, we have {a} s is a union of supra semi-closed sets.
Proof. 1 → 2: For each a = b ∈ X, there exists a supra semi-open set G containing a  but not b, or containing b but not a. Say a ∈ G and b ∈ G. Then a ∈ scl({b})  3 → 1: Let a = b. Then we have two cases: Since a ∈ {a} s , then a ∈ F . Therefore F c is a supra semi-open set containing a such that b ∈ F c .
(ii) Or b ∈ {a} s . Then there is a supra semi-open set G containing b such that a ∈ G.
In the both cases above, we infer that (X, µ) is an SsT 0 -space.  We show by the following example that the converse of the above proposition is not always true. We need the following definition to obtain the equivalence between SsT 0 and SsT 1 .
Definition 14. (X, µ) is is called a supra semi symmetric space if a ∈ scl{b} implies that b ∈ scl{a} for a = b ∈ X. Theorem 6. Let (X, µ) be a supra semi symmetric space. Then it is SsT 1 iff it is SsT 0 .
Proof. The necessary condition is obvious. To prove the sufficient condition, let a = b. Then there exist a supra semi-open set G containing only one of them. Say, a ∈ G and b ∈ G. Therefore a ∈ scl{b}. By the supra semi symmetry of (X, µ), we have b ∈ scl{a}. Thus (scl{a}) c is a supra semi-open set containing b. Hence, (X, µ) is SsT 1 .
Theorem 7. The following three statements are equivalent: a) : a ∈ X} is supra semi-closed in the product supra space X × X.
Proof. 1 → 2: Consider (X, µ) is an SsT 2 -space. Then for a = b, there exist two disjoint supra semi-open sets G i and H i such that a ∈ G i and b ∈ H i . Obviously, G i ⊆ H c i . Therefore a ∈ scl(G i ) ⊆ H c i = F i . Thus, F i is a supra semi-closed neighborhood of a such that b ∈ F i . Hence, {a} = {F i : F i is a supra semi-closed neighborhood of a}.
2 → 1: To prove that (X, µ) is an SsT 2 -space, let a = b. Since {a} = {F i : F i is a supra semi-closed neighborhood of a}, then there exists a supra semi-closed neighborhood F i 0 of a such that b ∈ F i 0 . Therefore there exists a supra semi-open set G containing a such that a ∈ scl(G) ⊆ F i 0 . It is clear that (scl(G)) c is a supra semi-open set containing b and G (scl(G)) c = ∅. Hence, (X, µ) is an SsT 2 -space.
1 → 3: Suppose that (X, µ) is SsT 2 and let (a, b) ∈ X × X − . Then a = b. Therefore there exist two disjoint supra semi-open sets G and H containing a and b, respectively. Thus, (a, b) ∈ G×H ⊆ X ×X − , proving that X ×X − is a supra semi neighbourhood of any of its points. Hence, is supra semi-closed. 3 → 1: Suppose that is a supra semi-closed subset of X × X and let a = b ∈ X. Then X × X − is a supra semi-open set containing (a, b). Therefore there exist two supra semi-open subsets G and H of (X, µ) such that (a, b) ∈ G × H ⊆ X × X − . This implies that G and H are two disjoint supra semi-open sets containing a and b, respectively. Hence, (X, µ) is SsT 2 .  (ii) (X, µ) is an SsT 1 -space; (iii) (X, µ) is an SsT 0 -space.
Proof. The implications 1 → 2 → 3 are obvious. 3 → 1: Let a, b ∈ X such that a = b. Since (X, µ) is an SsT 0 -space, then from Theorem (4), we get scl{a} = scl{b}. Therefore a ∈ scl{b} or b ∈ scl{a}. Say, a ∈ scl{b}. Since (X, µ) is supra semi regular, then there exist disjoint supra semi-open sets G and H containing a and scl{b}, respectively. Thus (X, µ) is an SsT 2 -space. (iii) For every supra semi-open sets U and V such that U V = X, there are two supra semi-closed sets F and H contained in U and V , respectively, such that F H = X.
Proof. 1 → 2: Consider (X, µ) is supra semi normal and F is a supra semi-closed subset of a supra semi-open set U . Then U c and F are disjoint supra semi-closed sets. Therefor there exist two disjoint supra semi-open sets W and V containing U c and F , respectively. Thus 2 → 3: Consider U and V are supra semi-open sets such that U V = X. Then U c is a supra semi-closed sets such that U c ⊆ V . By 2, there is a supra semi-open set G such that U c ⊆ G ⊆ scl(G) ⊆ V . Thus G c ⊆ U and scl(G) ⊆ V are supra semi-closed sets such that G c scl(G) = X. It should be noted that the concepts of ST i -space which were defined by replacing 'supra semi-open' by 'supra open' in Definition (13), see, [2,21].
Converse of this theorem is not necessary true as it is seen in the following examples. We complete this section by discussing these separation axioms in terms of hereditary and topological properties and finite product space.
Definition 15. For a nonempty subset A of (X, µ), the family µ A = {A G : G is a supra semi-open subset of (X, µ)} is called a relative semi-topology on A. A pair (A, µ A ) is called a semi-subspace of (X, µ).
One can easily prove that a semi-subspace (A, µ A ) of (X, µ) is a supra topological space.
Proof. Necessity: Let H be a supra semi-closed subset of (Y, µ Y ). Then there exists a supra semi-open subset W of (Y, µ Y ) such that H = Y \W . Now, there exists a supra semiopen subset V of (X, µ) such that W = Y V . Therefore H = Y \ (Y V ) = Y V c . By taking F = V c , the proof of the necessary part is complete.
Sufficiency: Let H = Y F such that F is a supra semi-closed subset of (X, µ). Then Since X \ F is a supra semi-open subset of (X, µ), then Y \ H is a supra semi-open subset of (Y, µ Y ). Thus H is a supra semi-closed subset of (Y, µ Y ).

Definition 16.
A property is said to be a relative semi-hereditary property if the property passes from a supra topological space to every relative semi-subspace.
Theorem 13. A property of being an SsT i -space is a relative semi-hereditary for i = 0, 1, 2, 3.
Proof. We shall suffice with proof of case i = 3 which directly contains the case i = 1. In a similar way, one can prove the cases i = 0, 2.
Suppose that (A, µ A ) is a relative semi-subspace of an SsT 3 -space (X, µ). We first show that (A, µ A ) is an SsT 1 -space. Let x = y ∈ A ⊆ X. Then there are two supra semi-open subsets U and V of (X, µ) containing x and y, respectively, such that x ∈ V and y ∈ U . Now, G = A U and H = A V are two supra semi-open subsets of (A, µ A ) containing x and y, respectively, such that x ∈ H and y ∈ G. Thus, (A, µ A ) is SsT 1 . Second, we show that (A, µ A ) is supra semi regular. Let H be a supra semi-closed subset of (A, µ A ) and a ∈ A such that a ∈ H. It follows from Proposition (5) that there is a supra semi-closed subset F of (X, µ) such that H = F A. Since a ∈ F , then there exist disjoint supra semi-open subsets U and V of (X, µ) containing F and a, respectively. Now, M = U A and N = V A are disjoint supra semi-open subsets of (A, µ A ) containing H and a, respectively. Thus (A, µ A ) is supra semi regular. Hence, the proof is complete. Proposition 6. Let g : (X, µ) → (Y, θ) be an injective supra semi-continuous map. If (Y, θ) is T i , then (X, µ) is SsT i for i = 0, 1, 2.
Proof. We only prove the proposition in the case of i = 2 and the other cases can be made similarly.
Let a = b ∈ X. Then, it follows from the injectivity of g, that there are x = y ∈ Y such that x = f (a) and y = f (b). Since (Y, θ) is T 2 , then there are two disjoint open subsets U and V of (Y, θ) containing a and b, respectively. Now, g −1 (U ) and g −1 (V ) are disjoint supra semi-open subsets of (X, µ) containing a and b, respectively. Hence, (X, µ) is SsT 2 , as required.
In a similar way, one can prove the following results. Let (X × Y, C) be the semi-product supra space of (X, µ) and (Y, ν). We first prove that (X × Y, C) is ST 1 . Suppose that (x 1 , y 1 ) = (x 2 , y 2 ). Then either x 1 = x 2 or y 1 = y 2 . Without loss of generality, suppose that x 1 = x 2 . Therefore there exist two supra semi-open subsets U and V of (X, µ) containing x 1 and x 2 , respectively. According to Definition (17), U × Y and V × Y are two supra open subsets of (X × Y, C) containing (x 1 , y 1 ) and (x 2 , y 2 ) such that (x 1 , y 1 ) ∈ V × Y and (x 2 , y 2 ) ∈ U × Y . Hence, (X × Y, C) is ST 1 . Second, we prove that (X × Y, C) is supra regular. Suppose that (x, y) ∈ X × Y and E is a supra closed subset of (X × Y, C) such that (x, y) where F i and H i are supra semi-closed subsets of (X, µ) and (Y, ν), respectively. Then there exists j ∈ I such that (x, y) ∈ [(F j × Y ) (X × H j )]. This means that x ∈ F j and y ∈ H j . Since (X, µ) and (Y, ν) are supra semi regular, then there exist disjoint supra semiopen subsets U and V of (X, µ) containing x and F j , respectively, and there exist disjoint supra semi-open subsets M and N of (Y, ν) containing y and H j , respectively. Therefore

Conclusion
There are many generalizations of topological spaces which help us to picture and satisfy many topological properties under fewer conditions such as supra topology, minimal structure and generalized topology. This work is devoted to introducing and discussing the concepts of limits points of a set and separation axioms with respect to semi-open sets. We have established their main properties and provided some examples to show the obtained results. From the concrete thoughts given in this work, it can be done more investigations on the theoretical parts of these generalized ideas which is valuable by studying the following themes: (i) Define weak types of supra semi regular and supra semi normal spaces. (iv) Investigate of the possibility of applying these concepts on information system, especially, separation axioms.