Complete Hausdorffness and Complete Regularity on Supra Topological Spaces

The supra topological topic is of great importance in preserving some topological properties under conditions weaker than topology and constructing a suitable framework to describe many real-life problems. Herein, we introduce the version of complete Hausdorffness and complete regularity on supra topological spaces and discuss their fundamental properties. We show the relationships between them with the help of examples. In general, we study them in terms of hereditary and topological properties and prove that they are closed under the finite product space. One of the issues we are interested in is showing the easiness and diversity of constructing examples that satisfy supra Ti spaces compared with their counterparts on general topology.


Introduction and Preliminaries
Topological space has been generalized in many manners. They can be classified into three main types; the first one was obtained by strengthening or weakening the conditions of a topology such as Alexendroff topology [1], supra topology [2], and generalized topology [3]. The second one was given by adding newly mathematical structures to the topology such as ordered topology [4], ideal topology [5], and bitopology [6]. The third one was established by defining a topology using one of the generalizations of crisp sets such as fuzzy topology [7] and soft topology [8]. Later on, wide generalizations were constructed by combining two or more types of the previous ones such as fuzzy bitopology [9], ideal ordered bitopology [10], and soft ordered topology [11].
Mashhour et al. [2] introduced the supra topology concept by deleting only the intersection condition. In analogy with topology, they studied the concepts of interior and closure operators, continuity, and separation axioms on supra topology. One of the concrete merits of supra topology is obtaining different examples that give meaning to the concepts and properties defined on a finite set; for example, the only topology defined on a finite set which is a T 1 space is the discrete topology, whereas there are several sorts of supra topologies that are a T 1 space. Additionally, it is a hard work to find counterexamples which illustrate that T i−ð1/2Þ spaces do not imply T i in the cases of i = 2ð1/2Þ, 3, 3ð1/2Þ, 4; see for example, [12,13]. After the emergence of supra topology, many authors have explored various topological notions in the spaces of supra topology such as compact [14,15] and paracompact spaces [16], neighborhood system [17], separation axioms [18]), operators, and generalized open sets [19,20]). Some published articles demonstrated that many topological findings are still true on supra topologies and illustrated that some of them are invalid like the distributive property of the closure (resp., interior) operator for the union (resp., intersection) between two sets and a compact set in a Hausdorff space is closed. In general, one can note that all topological properties and characterizations which are related to the intersection operator do not remain valid in supra topological spaces. In conclusion, we point to the existence of several studied extensions of supra topology such as supra ordered topology [21], supra fuzzy topology [22], supra soft topology [23], supra ideal topology [24], supra ordered bitopology [25], and supra soft ordered topology [26].
In the field of applications, the supra topology represents a framework that is general enough to model phenomena and handle practical issues. In [27], the authors elucidated how supra topological frames induced by semiopen sets handled some digital problems. It is well known that the classes of regular sets and semiopen sets define supra topology structures. In [28], these classes were examined to fix or eliminate obstacles on the digital scope.
Our goal in this article is to complete separation axioms reported in [19] by introducing the concepts of supra completely Hausdorff and supra completely regular spaces. We elaborate their master properties and demonstrate the relationships between them by some illustrative examples. Also, we investigate their behaviours with respect to subspaces, S å homeomorphism maps, and the finite product of spaces. Now, we mention some definitions and results in which we need to illustrate the obtained findings.
Definition 1 (see [2]). We call Σ as a supra topology on X provided that Σ is a subcollection Σ of 2 X such that X ∈ Σ and the arbitrary unions of members of Σ are a member of Σ. A pair ðX, ΣÞ is called a supra topological space. Every subset of X that belongs to Σ is called supra open, and a set is called supra closed if its complement belongs to Σ.
To refer supra topological spaces, we use the the pairs ðX, ΣÞ and ðY, ΩÞ.
Definition 2 (see [19]). For a subset E of ðX, ΣÞ, the union of all supra open sets that is contained in E is called the supra interior of E (denoted by sintðEÞ); the intersection of all supra closed sets that contains E is called the supra closure of E (denoted by sclðEÞ).
Definition 3 (see [2]). We call a map S å continuous if the inverse image of every supra open subset is a supra open subset.
Definition 5 (see [19]). A supra relative topology on a subset A of ðX, ΣÞ is given by Σ A = fA T G : G ∈ Σg. A pair ðA, Σ A Þ is said to be a supra subspace of ðX, ΣÞ.
Lemma 6 (see [19]). A subset E of ðA, Σ A Þ, which is a supra subspace of ðX, ΣÞ, is supra closed iff there is a supra closed subset F of ðX, ΣÞ such that E = A T F.
(1) ðX, ΣÞ is supra regular G i : G i ∈ Σ i g a base for a supra topology Σ on X = Q n i=1 X i . A pair ðX, ΣÞ is called a finite product supra space.

Supra Completely Hausdorff Spaces
In this portion, we define a concept of supra completely Hausdorff spaces and investigate its master properties. A number of examples which clarify relationships between this concept and some separation axioms that we consider is provided.
Definition 13. ðX, ΣÞ is said to be supra completely Hausdorff (or supra T 2ð1/2Þ ); if for any a ≠ b ∈ X, there are two supra open sets U and V containing a and b, respectively, such that sclðUÞ T sclðVÞ = ∅.

Corollary 15.
When ðX, ΣÞ is supra regular, the next concepts are identical.
(1) ðX, ΣÞ is supra T 0 The converse of the above result is not true in general as the two examples clarify as follows.
Example 16. We construct a supra topology Σ on X = fω 1 , ω 2 , ω 3 , ω 4 g as follows: It can be checked that ðX, ΣÞ is supra T 2ð1/2Þ . On the other hand, fω 1 , ω 4 g is a supra closed set and ω 2 ∈ fω 1 , ω 4 g. ðX, ΣÞ is not supra T 3 because there do not exist two disjoint supra open sets such that one of them includes ω 2 and the other includes fω 1 , ω 4 g.
Example 17. We construct a supra topology Σ on X = fa, b, x, y, zg as follows: x, y f gg: Obviously, ðX, ΣÞ is supra T 2 . In contrast, a ≠ b and the supra closure of any supra open set containing a and the supra closure of any supra open set containing b have a nonempty intersection, so that ðX, ΣÞ is not supra T 2ð1/2Þ . Theorem 18. The two concepts are identical if |X | ≤4 as follows.

Definition 22.
(1) A property that passes from a supra topological space to every supra subspace is called a supra hereditary property (2) A property that is preserved by any S å homeomorphism map is called a supra topological property  Proof. Consider ðX, ΣÞ and ðY, ΩÞ as two supra topological spaces.

Supra Completely Regular Spaces
In this portion, we first investigate some properties of supra regular and supra normal spaces. Then, we define a concept of supra completely regular spaces. Compared with completely regular on topological spaces, the continuous functions are replaced by S å continuous functions; however, the codomain remains ½0, 1 which is a subspace of the usual topology. We discuss some rudiments of a supra completely regular space with the help of examples and characterize it in many ways. The above five cases end the proof that ðX, ΣÞ is supra normal. (1) ðX, ΣÞ is supra T 4 (2) ðX, ΣÞ is supra T 3 By the next examples, we illustrate that the converse of Theorem 28 fails and elucidate that the concepts of supra normal and supra regular spaces are independent of each other when |X | >4.
Example 30. Consider that Σ = f∅,X, fag, fa, bg, fb, cgg is a supra topology on X = fa, b, cg. Obviously, ðX, ΣÞ is supra normal. In contrast, b ∈ fcg and fcg is a supra closed set. ðX, ΣÞ is not supra regular because there do not exist two disjoint supra open sets separate b and fcg.
Example 31. Consider that Σ = PðXÞ \ ffag, fxg, fa, bg, fx, ygg is a supra topology on X = fa, b, x, y, zg. Now, fa, bg and fx, yg are supra closed subsets of ðX, ΣÞ. One can check that ðX, ΣÞ is supra regular. In contrast, ðX, ΣÞ is not supra normal because there do not exist two disjoint supra open sets separate fa, bg and fx, yg.
The following result is the version of Urysohn's lemma on supra topology and has a similar proof to Urysohn's lemma on general topology, so that the proof is omitted. 4 Journal of Applied Mathematics Theorem 32. ðX, ΣÞ is supra normal iff for each disjoint supra-closed sets A, B there exists an S å continuous map g : X ⟶ ½0, 1 such that gðaÞ = 0 for each a ∈ A and gðbÞ = 1 for each b ∈ B.
Take into consideration that a supra topology which is defined on ½0, 1 is a subspace of the usual topological space ðR, ΣÞ.
Definition 33. ðX, ΣÞ is said to be (1) Supra completely regular if for each x ∈ X and supra closed set F such that if x ∈ F, there exists an S å continuous map g : X ⟶ ½0, 1 such that gðxÞ = 0 and gðFÞ = 1 (2) Supra T 3ð1/2Þ if it is supra completely regular and supra T 1 Proposition 34. The following three statements are identical: (1) ðX, ΣÞ is supra completely regular (2) For each x ∈ X and supra closed set F such that x ∈ F, there exists an S å continuous map g : X ⟶ ½0, 1 such that gðxÞ = 1 and gðFÞ = 0 (3) For each x ∈ X and supra open set U containing x, there exists an S å continuous map g : X ⟶ ½0, 1 such that gðxÞ = 1 and gðU c Þ = 0 Proof. 1 ⟶ 2. Since ðX, ΣÞ is supra completely regular, then for each x ∈ X and supra closed set F such that x ∈ F, there exists an S å continuous map g : X ⟶ ½0, 1 such that gðxÞ = 0 and gðFÞ = 1. Therefore, a map hðxÞ = 1 − gðxÞ of ðX, ΣÞ into ½0, 1 is S å continuous. Hence, hðxÞ = 1 and hðFÞ = 0, as required. 2 ⟶ 3, obviously. 3 ⟶ 1. Let F be a supra closed set and x ∈ X such that x ∈ F. Then, F c is a supra open set. By hypothesis, there exists an S å continuous map g : X ⟶ ½0, 1 such that gðxÞ = 1 and gððF c Þ c Þ = gðFÞ = 0. Therefore, a map hðxÞ = 1 − g ðxÞ of ðX, ΣÞ into ½0, 1 is S å continuous. Since hðxÞ = 0 and hðFÞ = 1, then, ðX, ΣÞ is supra completely regular.
Theorem 35. For every distinct points a, b in a supra T 3ð1/2Þ space ðX, ΣÞ, there is an S å continuous function f : ðX, ΣÞ ⟶ R such that f ðaÞ ≠ f ðbÞ.
Definition 36. For an S å continuous map g of X into R, we define (1) g + of X into R by g + ðxÞ = sup fgðxÞ, 0g. It is obvious that g + is S å continuous (2) The zero set ZðgÞ by ZðgÞ = fx ∈ X : gðxÞ = 0g Theorem 37. ðX, ΣÞ is supra completely regular iff for every x ∈ X and supra closed set F such that x ∈ F, there are S å continuous maps f and g of ðX, ΣÞ into R such that x ∈ sin t½Zð f Þ, F ⊆ sin t½ZðgÞ, and Zð f Þ T ZðgÞ = ∅. Proof.
(1) Necessity. Suppose that F is a supra closed set and x ∈ X such that x ∈ F. (2) Sufficiency. Let F be a supra closed set and x ∈ X such that x ∈ F. Take two maps f and g as described in the theorem. Then, we have f 2 ðxÞ + g 2 ðxÞ ≠ 0 for each x ∈ X. Therefore, hðxÞ = f 2 ðxÞ½ f 2 ðxÞ + g 2 ðxÞ −1 is an S å continuous map such that hðxÞ = 0 and hðFÞ = 1. Hence, ðX, ΣÞ is supra completely regular Theorem 38. Every supra completely regular space ðX, ΣÞ is supra regular.
Proof. Let F be a supra closed subset of ðX, ΣÞ and let x ∈ F. Since ðX, ΣÞ is supra completely regular, there is an S å continuous map g : X ⟶ ½0, 1 such that gðxÞ = 0 and gðFÞ = 1. Since ½0, 1 is a subspace of the usual topological space ðR, ΣÞ, it is supra T 2 . So that there are two disjoint supra open sets U and V containing 0 and 1, respectively. It follows from the S å continuity of g that g −1 ðUÞ and g −1 ðVÞ are two disjoint supra open sets containing x and F, respectively. Hence, ðX, ΣÞ is supra regular.
Theorem 40. A supra normal space ðX, ΣÞ is supra completely regular iff it is supra regular. Proof.
(1) Necessity. It follows from the above theorem (2) Sufficiency. Let F be a supra closed subset of ðX, ΣÞ and let x ∈ F. By hypothesis, there are two disjoint supra open sets U and V such that x ∈ U and F ⊆ V. Now, F and V c are two disjoint supra closed sets. By using Theorem 32, we can find an S å continuous 5 Journal of Applied Mathematics map g : X ⟶ ½0, 1 such that gðaÞ = 0 for each a ∈ V c and gðbÞ = 1 for each b ∈ F. This means that gðxÞ = 0 and gðbÞ = 1 for each b ∈ F. Hence, ðX, ΣÞ is supra completely regular.
Corollary 41. Every supra T 4 space ðX, ΣÞ is supra T 3ð1/2Þ . Proof. Consider that ðA, Σ A Þ is a supra subspace of a supra completely regular space ðX, ΣÞ. Let E be a supra closed subset of ðA, Σ A Þ and x ∈ A such that x ∈ E. Then, it follows from Lemma 6 that there is a supra closed subset F of ðX, Obviously, x ∈ F. By the supra complete regularity of ðX, ΣÞ, there is an S å continuous map g : X ⟶ ½0, 1 such that gðxÞ = 0 and gðFÞ = 1. Thus, g | A : ðA, Σ A Þ ⟶ ½0, 1 is an S å continuous map such that g | AðxÞ = 0 and g | AðEÞ = g | AðA T FÞ = 1. This ends the proof that a supra completely regular space is a supra hereditary property.
Corollary 44. A supra T 3ð1/2Þ space is a supra hereditary property.
Theorem 45. A supra completely regular space is a supra topological property.
Corollary 46. A supra T 3ð1/2Þ space is a supra topological property.
Theorem 47. The finite product of supra completely regular spaces is supra completely regular.
Proof. Let ðX × Y, Σ × ΩÞ be the product spaces of two supra completely regular spaces ðX, ΣÞ and ðY, ΩÞ. Suppose that ðx, yÞ ∈ X × Y and G = S i∈I ðU i × V i Þ be a supra open subset of X × Y such that ðx, yÞ ∈ G. Now, there exists α ∈ I such that ðx, yÞ ∈ U α × V α . Then, for each x ∈ X and supra open set U α containing x, there exists an S å continuous map g 1 : X ⟶ ½0, 1 such that g 1 ðxÞ = 1 and g 1 ðU c α Þ = 0, and for each y ∈ Y and supra open set V α containing y, there exists an S å continuous map g 2 : X ⟶ ½0, 1 such that g 2 ðyÞ = 1 and g 2 ðV c α Þ = 0. Now, we define a map f : X × Y ⟶ ½0, 1 by f ðu, vÞ = g 1 ðuÞ × g 2 ðvÞ. It can be checked that f is S å continuous, we have f ðx, yÞ = g 1 ðxÞ × g 2 ðyÞ = 1 × 1 = 1 and Corollary 48. The property of being a supra T 3ð1/2Þ space is preserved under a finite product.

Conclusion
In this paper, we have highlighted two concepts on supra topological spaces, namely, supra completely Hausdorff and supra completely regular spaces. We have investigated some of their characterizations and provided some examples to show the relationships between them. Also, we have proved that they are hereditary and topological properties as well as they are closed under the finite product of spaces.
In future works, we plan to study an application of supra topology on the information system and apply the concepts given herein and those in [19] to categorize the approximation spaces. Also, we will study functionally separation axioms [29] in the frame of supra topologies and elaborate their relationships with the concepts presented herein.

Data Availability
No data were used to support this study.

Conflicts of Interest
The author declares that he has no competing interests.