On Some Properties of Whyburn Spaces

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Introduction
Alexandroff space is a topological space such that its collection of open sets is closed under arbitrary intersection. In 1937, Alexandroff introduced these spaces with the name of "Diskrete Räume" [1]. In [2], Steiner has named them principal spaces. Alexandroff spaces are used and applied in different domains like geometry, theoretical physics, and diverse branches of computer sciences. After that, Alexandroff spaces played an important role in digital topology (cofinite spaces) (see [3][4][5][6]). e specialization quasiorder of an Alexandroff space is defined by Now, if ≤ is a quasiorder on space X then the set of all supersets B: � x↑: x ∈ X { } (x↑: � y ∈ X: x ≤ y .) forms a basis of an Alexandroff topology Y( ≤ ) on X. In this case, the closure x { } is exactly the downset x↓: � y ∈ X: y ≤ x . We denote by v(x) � x↑ the minimal neighborhood of x. For more information on Alexandroff spaces you can see [7][8][9][10][11][12][13].
In [14], Echi introduced a particular class of Alexandroff spaces named primal spaces. (X, τ) is called a primal space if there exists a map f: X ⟶ X such that τ � Y(f), where Y(f) is the collection of all f-invariant subsets of X (for more information see [14,15]). In [16], the authors characterized maps f such that the primal space (X, Y(f)) is submaximal or door.
is paper is devoted to characterizing Alexandroff spaces which are submaximal, door, n-resolvable, Whyburn, and weakly Whyburn. Some useful examples are presented and commented and finally, all results on primal spaces are deduced. In the first section of this paper, we will give characterizations of Alexandroff spaces to be a submaximal, door, and n-resolvable. e second section is devoted to introducing and characterizing topological spaces, called quasi-Whyburn spaces, such that their T 0 -refections are Whyburn. Particularly, the case of Alexandroff spaces is totally deduced in this particular class of spaces.

Submaximal, Door, and n-Resolvable Alexandroff Spaces
We know that a submaximal space is a topological space in which all dense subsets are open. e following theorem characterizes submaximal spaces [17].
We can find the specialization quasiorder as follows: We have (X, τ) is an Alexandroff space which is submaximal.
(2) Consider the set N of natural numbers and for each (1) X is submaximal (2) e specialization quasiorder is an order and every chain in its graph has a length less than or equal to 2 Suppose that X is submaximal. If x, y are two elements of X satisfying x ≤ y and y ≤ x then x ∈ y and y ∈ x, which imply that x { } � y . Using the fact that every submaximal space is T 0 , we deduce that x � y and then the specialization quasiorder is an order. Now, suppose that x < y < z. If Y � x, z { }, then y ∈ Y∖Y and x ∈y⊆Y∖Y. Since x ∉ Y∖Y then it is not closed.
Using 2), for every a ∈ Y∖Y c and for any so that Y∖Y is closed and then X is submaximal. □ Corollary 1 (see [14], eorem 4.1) Hence, every subset of X not containing 0 is open. Yet, every subset of X containing 0 is closed. erefore, X is an Alexandroff door space. Considering Alexandroff door spaces, the second main result of this section is given by the following theorem. But, first, we need to recall increasing and decreasing sets. e increasing hull of a set A in a quasiordered set  □ Corollary 2 (see [16], Now, we will give a study of n − resolvable Alexandroff spaces. First, let us recall the definition of n − resolvable spaces. If X is a topological space, then it is called n − resolvable (n > 1) if there exist n-many mutually disjoint dense sets of X. A 2-resolvable space is called a resolvable space. Hewitt added also the condition "has no isolated points" to the definition of resolvable spaces. Also, a topological space is n − resolvable if and only if it is the union of n-many mutually disjoint dense subsets.
Stone [18] characterizes Alexandroff spaces which are n-resolvable in the following theorem.
Theorem 4 (see [18]). Let (X, ≤ ) be a quasiordered set. en, we have equivalence between the following statements: (1) X admit a partition into n mutually disjoint cofinal sets (2) ∀x ∈ X, x↑ has at least n elements We note that, for every subset Y of an Alexandroff space X, we have equivalence between the following items: is allows us to rephrase Stone's result as follows.
Theorem 5. Let (X, τ) be an Alexandroff space and ≤ is its specialization quasiorder. en, the following statements are equivalent:

has no isolated points
We recall that the T 0 -reflection of a topological space X is the quotient space denoted by T 0 (X): � X/ ∼ obtained from the equivalence relation defined on X by x ∼ y if and only if x { } � y . (1) X is n-resolvable (2) ∀x ∈ X, x↑ contains at least n distinct points (3) Every maximal element x in the T 0 -refection T 0 (X) arises from a cycle x � x 1 < x 2 < x 3 < · · · < x n < x n + 1 � x containing at least n distinct points x i Now, we shed some light on interesting examples.

Example 2
(1) Consider the set Z of all integers with the usual order ≤ . For any integer n ≥ 2 let A k � j ∈ Z: j � k mod n . en A 0 , A 1 , . . . , A n−1 are mutually disjoint dense sets of the Alexandroff space (Z, Y( ≤ )), showing that this space is n-resolvable for every n ≥ 2. Indeed, it is obvious that n↑ is infinite for each n ∈ Z.
(2) Let ≤ be the inverse order of N where N is the set of all natural numbers. In the Alexandroff space (N, Y( ≤ )), every set A of X is dense if and only if 0 ∈ A. erefore, (N, Y( ≤ )) is not a resolvable space. In fact, we note that |0↑| � 1.

Alexandroff Spaces Which Are Whyburn and Quasi-Whyburn Spaces
Let X be a topological space and F be a subset of X. en, F is called almost closed if and only if F∖F � x { } for some x ∈ X. We use the notation F ⟶ x. e notion of Whyburn spaces was first introduced as accessibility spaces by G.T. Whyburn in his famous paper [19]. Hence, a Whyburn space is a topological space X satisfying Y⊆X, A topological space X is called weakly Whyburn [20] if We denote the class of all Whyburn spaces (resp., weakly Whyburn spaces) by AP-spaces (resp., WAP-spaces) [21][22][23].

Quasi-Whyburn Spaces. A continuous map q from a topological space X to a topological space Y is said to be a quasihomeomorphism if U↦q − 1 (U) defines a bijection between the collection of all open sets of Y and the collection of all open sets of X [24].
We can see easily that the canonical surjection μ X : X ⟶ T 0 (X) is a quasihomeomorphism. More precisely, μ X is an onto quasihomeomorphism, and in this case, the following results are useful.

Lemma 1 (see [25]). Let μ: X ⟶ Y be continuous onto the map. en, μ is a quasihomeomorphism if and only if μ is an open map and q − 1 (q(A)) � A for every open subset A of X;
equivalently, μ is a closed map and q − 1 (q(A)) � A for every closed subset A of X.

Lemma 2 (see [16]). A quasihomeomorphism μ: X ⟶ Y is onto if and only if μ
If X is a topological space, x ∈ X, and Y⊆X, we take the notations in [16]. In that paper, authors denote by d 0 (x) the subset y ∈ X: x { } � y and by d 0 (Y) the union of d 0 (a) for all a ∈ Y.
Using these notations, we can find the following properties: Now, we introduce the notions of d 0 -closed subsets, in a given topological space, and quasi-Whyburn spaces as follows.

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Definition 2. Let A be a subset of a topological space X. en, called a quasi-Whyburn space or a QAP-space (or also a T 0 -Whyburn space) e following theorem gives a characterization of quasi-Whyburn spaces. Theorem 6. If X is a topological space, then we have equivalence between the following statements: (2) ⟹ (1). Conversely, let A⊆X such that μ X (A) is not closed in T 0 (X) and consider a point x in X with μ X (x) ∈ μ X (A)\μ X (A).
en, x ∈ A\d 0 (A) with A nond 0 -closed. So, by hypothesis, there is a subset B of X such □ Definition 3. A topological space X is called quasiweakly Whyburn space (or T 0 -weakly Whyburn space) and denoted by QWAP-space if its T 0 -reflection is a weakly Whyburn space. e proof of the following result is similar to that of eorem 6. Theorem 7. If X is a topological space, then we have equivalence between the following statements: of X with d 0 (B)⊆d 0 (A) and B\d 0 (A) � d 0 (x), for some x ∈ X

Alexandroff Spaces Which Are Whyburn and Quasi-Whyburn Spaces
Theorem 8. If X is an Alexandroff space, then we have equivalence between the following statements: Proof. Suppose that X is Whyburn and there exists x ∈ X such that (↓x)∖ x { } contains two distinct elements y and z.
which leads to a contradiction because (↓x)∖ x { } contains also z. Conversely, suppose that each element of X has at most 2 predecessors.
Let Y⊆X such that Y ≠ Y. Using the fact that X is Alexandroff, we have Let x ∈ Y∖Y and t ∈ Y satisfying x ∈↓t. Since |↓t| ≤ 2, en, we have equivalence between the following statements: Clearly, for any nonempty subset A of X, we have A � A ∪ ∞ { }; then, the condition x ∈ A\A means that x � ∞, and thus, A ⟶ x. We observe that, in this case, for every n ∈ Z, |↓n| � | n, ∞ { }| � 2 and |↓∞| � | ∞ { }| � 1. One can illustrate this situation in Figure 1.
Proof. It is enough to see that an Alexandroff Whyburn space is a functionally Alexandroff space. Hence, by eorem 8, |↓x| > 2 for any x ∈ X. Two cases arise as follows: (a) If |↓x| � 1, we take f(x) � x, and thus x If |↓x| � 2, then ↓x � x, y , and in this case, we take Proof. e first remark that X is Alexandroff if and only if T 0 (X) is Alexandroff and for any x, y ∈ X; we have x ≤ y⟺μ X (x) ≤ μ X (y).
T 0 (X) is defined by Figure 5which is not a functionally Alexandroff space.
X is a QWAP − space⟺T 0 (X) is a WAP − space ⟺T 0 (X) is a AP − space ⟺X is a QAP − space.
□ Data Availability e data set can be accessed upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.  Computational Intelligence and Neuroscience