Some Applications of Supra Preopen Sets

,e aim of this work is to define some concepts on supra topological spaces using supra preopen sets and investigate main properties. We started this paper by correcting some results obtained in previous study and presenting further properties of supra preopen sets.,en, we introduce a concept of supra prehomeomorphismmaps and discuss its main properties. After that we explore the concepts of supra limit and supra boundary points of a set with respect to supra preopen sets and examine their behaviours on the spaces that possess the difference property. Finally, we formulate the concepts of supra pre-Ti-spaces (i � 0, 1, 2, 3, 4) and give completely descriptions for each one of them. In general, we study their main properties in detail and show the implications of these separation axioms among themselves as well as with STi-space with the help of some interesting examples.


Introduction and Preliminaries
A set X with a family μ of its subsets is called a supra topological space [1], denoted by (X, μ), if X ∈ μ and the arbitrary union of members of μ is in μ. Mashhour et al. [1] generalized some topological notions such as interior and closure operators and continuity and separation axioms. Al-Shami [2] has studied the classical topological notions such as limit points of a set, compactness, and separation axioms on the supra topological spaces.
Some results via topology are 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 and is invalid on the supra topologies. To extend a class of supra open sets, the notions of supra α-open [3], supra preopen [4], supra b-open [5], supra β-open [6], supra R-open [7], and supra semiopen sets [8] have been introduced and their main properties have been discussed. ese 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. ese generalizations have been utilized to define new versions of compactness and connectedness, see, for example [9][10][11][12][13][14]. Mustafa and Qoqazeh [15] took advantage of supra D-sets to define separation axioms on supra topological spaces. Recently, Al-Shami and El-Shafei [16] have studied separation axioms on supra soft 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 [17]. Also, the possibility of applying semiopen sets to deal with some problems on digital topology has been demonstrated in [18]. e layout of the paper is as follows. In Section 2, we correct some results of [4] and investigate further properties of supra preopen sets. Also, it presents the concept of supra prehomeomorphism maps and explores main properties. e concepts of supra prelimit and supra preboundary points of a set are studied in Section 3. Section 4 introduces new types of separation axioms using supra preopen sets and elucidates the relationships between them with the help of examples. Section 5 concludes the paper with summary and further works.
In what follows, we collect the relevant definitions and results from supra topology and supra preopen sets to make this paper self-contained and easy to read.
Definition 1 (see [1]). A family μ of subsets of a nonempty set X is called a supra topology provided that the following two conditions hold: (1) X and ∅∈ μ (2) μ is closed under arbitrary union en, 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
(1) Since ∪ i∈∅ G i � ∅, then some authors remove the empty set ∅ from the first condition of a supra topology (2) μ is called an associated supra topology with a topology τ if τ ⊆ μ (3) rough this paper, we consider (X, μ) and (Y, ]) are associated supra topological spaces with the topological spaces (X, τ) and (Y, θ), respectively Definition 2 (see [1]). Let A be a subset of (X, μ). en, 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. [3] if A⊆int(cl(int(A))) (2) Supra preopen [4] if A⊆int(cl(A)) (3) Supra b-open [5] if A⊆int(cl(A))∪cl(int(A)) Definition 4 (see [1]). For a subset A of (X, μ), pint μ (A) is the union of all supra preopen sets contained in A and pcl μ (A) is the intersection of all supra preclosed sets containing A.
If there is no confusion, we write pint(A) and pcl(A) in the places of pint μ (A) and pcl μ (A), respectively.
A pair (A, μ A ) is called a supra subspace of (X, μ).
Definition 8 (see [11]). 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.
Definition 9 (see [11]). Let (X i , μ i ): i � 1, 2, . . . , n be the collection of supra topological spaces. en, β � n i�1 μ i � n i�1 G i : G i ∈ μ i defines a basis for a supra topology T on X � n i�1 X i . e pair (X, T) is called a finite product supra spaces.
We give the below example to demonstrate that the above proposition need not be true in general.
} be a supra topology on X � 1, 2, 3, 4 { }. Since int(cl( 2, 4 { })) � X, then 2, 4 { } is a supra preopen set and since int(cl(int Since int(cl ( 4 { })) � ∅ and int(cl(int( 4 { }))) � ∅, then 4 { } is neither a supra preopen set nor a supra α-open set. is emphasizes that the above proposition is erroneous. e above proposition is true in the case of (X, μ) is a topological space because the following two properties are satisfied on topological spaces, but they do not valid in supra topological spaces: . us, A is a supra preopen set. Also, (int(A)) c � X. is implies that cl(A c ) � X. erefore, A c is supra preopen set.

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In general, there does not exist a relationship between supra preopen sets in (X, μ) and its subspaces as the next two example show.
us, g(A)⊆int ] (cl ] (g(A))). Hence, g(A) is a supra preopen set. □ Definition 10. For a nonempty subset A of (X, μ), the family μ A � A∩G: G { is a supra preopen subset of(X, μ)} is called a relative pretopology on A. A pair (A, μ A ) is called a presubspace of (X, μ).
One can easily prove that a presubspace (A, μ A ) of (X, μ) is a supra topological space.
en, there exists a supra preopen subset W of (Y, μ Y ) such that H � Y\W. Now, there exists a supra preopen subset V of (X, μ) such that W � Y∩V. erefore, Since X\F is a supra preopen subset of (X, μ), then Y\H is a supra preopen subset of (Y, μ Y ). us, H is a supra preclosed subset of (Y, μ Y ).

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In the rest part of this section, we present a concept of supra prehomeomorphism and supra pre ⋆ -homeomorphism maps and discuss some basic properties ( ⋆ denotes another type of homeomorphism maps).
is said to be a supra prehomeomorphism if it is supra precontinuous and supra preopen.
Since every supra open set is supra preopen, then every supra homeomorphism map is a supra prehomeomorphism. However, the converse is not always true as it is illustrated in the following example.
} are two topologies on the real numbers set R.
{ }, R} be two associated supra topologies with τ and θ, respectively. en, the identity map g: (R, μ) ⟶ (R, ]) is a supra prehomeomorphism, but it is not a supra homeomorphism because the image of a supra open set 1, 3 { } is on a supra open set.

Theorem 5.
e equivalence of the following properties hold if a map g: (X, μ) ⟶ (Y, ]) is bijective and supra precontinuous: is a supra prehomeomorphism if and only if pcl ] (g(A))⊆g(cl τ (A)) and g(pcl μ (A))⊆cl θ (g(A)) for every A⊆X.
′ ⟹ ′ If g is a bijective map such that pcl ] (g (A))⊆g(cl τ (A)) and g(pcl μ (A))⊆cl θ (g(A)), then g is supra precontinuous and supra preclosed. It follows from eorem 1 that g is a supra prehomeomorphism map.
is a supra preopen (resp. supra preclosed) set in Y for every supra preopen (resp. supra preclosed) set in X (3) Supra pre ⋆ -homeomorphism if it is bijective, supra pre ⋆ -continuous, and supra pre ⋆ -open

Limit and Boundary Points of a Set with Respect to Supra Preopen Sets
is section defines the concepts of supra prelimit and supra preboundary points of a set and studies the interrelations between them. It provides some examples to show the obtained results and examines some properties of the supra prederived set on the spaces that possess the difference property.
Definition 13. A subset A of (X, μ) is said to be a supra preneighbourhood of x ∈ X provided that there is a supra preopen set F containing x such that x ∈ F⊆A.

Definition 14.
A point x ∈ X is said to be a supra prelimit point of a subset A of (X, μ) provided that every supra preneighborhood of x contains at least one point of A other than x itself.
All supra prelimit points of A is said to be a supra prederived set of A and is denoted by A p′ .

Proposition 4.
If A⊆B, then A p′ ⊆B p′ for every subsets A and B of (X, μ).
Proof. Straightforward. □ Corollary 1. We have the following results for any two subsets A and B of (X, μ): e following example illustrates that the converse of the above proposition and corollary fails.
Obviously, X p′ � X. Now, we have the following cases: Proof Necessity: let x ∈ A p′ . en, for every supra preopen set Let A be a subset of (X, μ). en, the following results hold.
en, A c is a supra preopen set containing x. In this case, A c ∩A � ∅ leads to x ∉ A p′ . erefore, A p′ ⊆A. Conversely, let x ∈ A c and let A p′ ⊆A. en, x ∉ A p′ . erefore, there is a supra preopen set G x such that From (1) and (2), we obtain G∩(A∪A p′ ) � ∅. is implies that x ∉ (A∪A p′ ) p′ . Hence, (A∪A p′ ) p′ ⊆(A∪A p′ ). By (i), A∪A p′ is a supra preclosed set, as required. Proof. Let a ∉ (g(A)) p′ . en, there is a supra preopen set H containing a such that (H\ a us, g − 1 (a) ∉ A p′ . Since g is bijective, then a ∉ g(A p′ ). erefore, g(A p′ )⊆(g(A)) p′ . By reversing the preceding steps, we find that (g(A)) p′ ⊆g(A p′ ). Hence, the proof is complete.   (N, μ) has the difference property. Also, it can be seen that the collection of supra preopen subsets of (N, μ) coincides with the collection of supra open sets. Hence, (N, μ) has the difference property for the collection of supra preopen sets. Theorem 9. If (X, μ) has the difference property for the collection of supra preopen sets, then the following properties hold for A⊆X: Since (X, μ) has the difference property for the collection of supra preopen sets, then G\ x { } is a supra preopen set.
(2) Since (A p′ ) p′ ⊆A p′ , then it follows from eorem 7 that A p′ is a supra preclosed set. erefore, Also, (A) p′ ⊆(pcl(A)) p′ because A⊆pcl(A). On the contrary, let x ∉ (A) p′ . en, it follows from 1 above From (3) and (4), the desired result is proved. (3) Let A be a finite subset of X. Suppose that there exists an element x ∈ X such that x ∈ A p′ . en, for every supra preopen set G containing x, we have G\ x { }∩A ≠ ∅. erefore, for every y ∈ A such that y ≠ x, we have G\ x, y is a supra preopen set. us,

G\[A∪ x
{ }] is a supra preopen set such that is implies that x ∉ A p′ . However, this is a contradiction. Hence, it must be that A p′ � ∅.

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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 preopen sets. Let A � 2, 4, 5 { } be a subset of supra topological space given in Example 6. Note that the collection of supra open sets coincides with the collection of supra preopen sets. By calculating, we find that A p′ � N\ 2, 4 { }, (A p′ ) p′ � N\ 1, 3 { }, and cl(A p′ ) � N. is leads to the following three properties: Definition 16. Let A be a subset of (X, μ). e supra preboundary of A (denoted by pb(A)) is the set of all elements which belongs to (pint(A)∪pint(A c )) c .

Lemma 1. Let
A be a subset of (X, μ). en, Proof. We prove (1) and (2) is proved in a similar way.

1) A is a supra preopen set iff pb(A)∩A � ∅ (2) A is a supra preclosed set iff pb(A)⊆A
Proof.
(2) Necessity: since A is supra preclosed, then is implies that x ∈ G⊆A c . us, A c is a union of supra preopen sets. Hence, A c is supra preopen. □ Corollary 4. We have the following properties for a subset A of (X, μ): (1) A is both supra preopen and supra preclosed iff

Separation Axioms with Respect to Supra Preopen Sets
In this section, we use a class of supra preopen sets to introduce new types of separation axioms, namely, there exist two disjoint supra preopen sets U and V containing a and b, respectively (4) Supra preregular if for every supra preclosed set F and each a ∉ F, there exist disjoint supra preopen sets U and V containing F and a, respectively (5) Supra prenormal if for every disjoint supra preclosed sets F and H, there exist disjoint supra preopen sets U and V containing F and H, respectively (6) SpT 3 (resp. SpT 4 ) if it is both supra preregular (resp. supra prenormal) and SpT 1 Theorem 10. e following three statements are equivalent:

union of supra preclosed sets
Proof. 1 ⟶ 2: for each a ≠ b ∈ X, there exists a supra preopen set G containing a but not b or containing b but not  a. Say a ∈ G and b ∉ G. en, a ∉ pcl( . en, we have two cases: Journal of Mathematics erefore, F c is a supra preopen set containing a such that b ∉ F c . (ii) Or b ∉ a { } p′ . en, there is a supra preopen set G containing b such that a ∉ G.
In the both cases above, we infer that (X, μ) is an SpT 0 -space. Proof. Let (X, μ) be an SpT 0 -space. Suppose that there are two distinct singleton set a is not an SpT 0 -space, a contradiction. Hence, (X, μ) contains at most a supra predense singleton set. □ Theorem 11.
e following there statements are equivalent:

) Every singleton subset of (X, μ) is supra preclosed (3) e intersection of all supra preopen sets containing a set A is exactly
is implies that any supra preopen set containing b contains a as well. us, the intersection of all supra preopen sets containing b is not equal b { }. However, this contradicts 3. Hence, it must be a  Proof. Let a ≠ b ∈ X. Since X is a supra preopen set and (X, μ) satisfies the difference property for the collection of supra preopen sets, then X\ a { } and X\ b { } are supra preopen sets containing b and a, respectively, such that a ∉ X\ a 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 SpT 0 and SpT 1 .
(a, b) ∈ G× H⊆X × X − Δ. is implies that G and H are two disjoint supra preopen sets containing a and b, respectively. Hence, (X, μ) is SpT 2 . □ Theorem 14. e following three statements are equivalent: (1) (X, μ) is supra preregular (2) For each supra preopen subset U of (X, μ) containing a, there exists a supra preopen subset V of (X, μ) such that a ∈ V⊆pcl(V)⊆U (3) Every supra preopen subset U of (X, μ) can be represented as follows: U � ∪ H: H { is a supra preopen subset of (X, μ) and pcl(H)⊆U} Proof. 1 ⟶ 2: let (X, μ) be a supra preregular space and U be a supra preopen set such that a ∈ U. en, there exist disjoint supra preopen sets V and W containing a and U c , respectively. erefore, a ∈ V⊆W c ⊆U. us, a ∈ V⊆ pcl(V)⊆U.
2 ⟶ 3: suppose that U is a supra preopen set. By hypothesis, for each a ∈ U, there exists a supra preopen set H such that a ∈ H⊆pcl(H)⊆U.
3 ⟶ 1: let F be a supra preclosed set such that a ∉ F. en, F c � ∪ H: H { is supra preopen and pcl(H)⊆F c }. Since a ∈ F c , then there exists a supra preopen set H a containing a such that pcl(H a )⊆F c . Take V � (pcl(H a )) c . en, V is a supra preopen set containing F and V∩H a � ∅.
is completes the proof. □ Theorem 15. Consider (X, μ) is a supra preregular space. en, the following concepts are equivalent: 2 ⟶ 3: consider U and V are supra preopen sets such that U∪V � X. en, U c is a supra preclosed sets such that U c ⊆V. By 2, there is a supra preopen set G such that U c ⊆G⊆pcl(G)⊆V.
us, G c ⊆U and pcl(G)⊆V are supra preclosed sets such that G c ∪pcl(G) � X.
3  Proof. Let a ∈ X such that cl( a { }) � A⊆X. By hypothesis, A is a supra clopen set.
{ } is a supra preopen set. e arbitrary selection of x implies that every singleton subset of X is supra preopen. erefore, every subset of X is supra preopen. Hence, the desired result is proved.

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To explain that the converse of the above theorem is not always true, we provide the following example.
} be a supra topology on X � 1, 2, 3 { }. en, the collection of all supra preopen subsets of (X, μ) is { }}. It easily checks that (X, μ) is an SpT 4 -space. On the contrary, a set 2, 3 { } ∈ μ is not a supra clopen set. Now, we show the implications of these separation axioms among themselves as well as with ST i -space.
It should be noted that the concepts of ST i -space which were defined by replacing "supra preopen" by "supra open" in Definition 4.1, see [1,2].
Converse of this theorem is not necessary true as seen from the following examples.  erefore, (X, μ) is not an SpT 1 -space because 1 ≠ 4 and every supra preopen set containing 1 contains 4 as well. On the contrary, it can be checked that (X, μ) is SpT 0 .
Example 10. Assume that (X, μ) is the same as in Example 7. en, (X, μ) is not an SpT 2 -space because 2 ≠ 3 and there do not exist disjoint supra preopen sets such that one of them and H ∈ C 2 } defines a basis for a supra topology C on X × Y. We called (X × Y, C) a prefinite product supra space.

Lemma 2.
Let (X, μ) and (Y, ]) be two supra topological spaces and (X × Y, C) be their preproduct supra space. If E is a supra closed subset of (X × Y, C), , where F i and H i are supra preclosed subsets of (X, μ) and (Y, ]), respectively.
Proof. We prove the theorem for two supra topological spaces (X, μ) and (Y, ]) in the case of i � 3. One can prove the other cases similarly.
en, either x 1 ≠ x 2 or y 1 ≠ y 2 . Without loss of generality, suppose that x 1 ≠ x 2 . erefore, there exist two supra preopen subsets U and V of (X, μ) containing x 1 and x 2 , respectively. According to Definition 20, 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 preclosed subsets of (X, μ) and (Y, ]), respectively. en, there exists j ∈ I such that (x, y) ∉ [(F j × Y)∪(X × H j )]. is means that x ∉ F j and y ∉ H j . Since (X, μ) and (Y, ]) are supra pre regular, then there exist disjoint supra preopen subsets U and V of (X, μ) containing x and F j , respectively, and there exist disjoint supra preopen subsets M and N of (Y, ]) containing y and us, (X × Y, T) is supra regular. Hence, the proof is complete.

Conclusion
We began this work by correcting some results of [4]. en, we have presented the concept of supra pre homeomorphism and investigated main properties. Also, we have introduced and studied the concepts of supra limits and supra boundary points with respect to preopen sets. Finally, we have defined the concepts of supra preregular, supra prenormal, and SpT i -spaces (i � 0, 1, 2, 3, 4) and discussed their basic properties. From the concrete thoughts given in this work, more investigations can be carried out on the theoretical parts of these generalized ideas which are valuable by studying the following themes: (1) Define weak types of supra preregular and supra prenormal spaces (2) Study SpT i -spaces for 1/2, 2 1/2, 3 1/2, 5 (3) Explore the concepts introduced herein using the classes of supra α-open sets, supra semiopen sets, supra b-open sets, and supra β-open sets (4) Investigate of the possibility of applying these concepts on information system, especially, separation axioms

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.