Quasihomeomorphisms and Some Topological Properties

: In this paper, we study the properties of topological spaces preserved by quasihomeomor-phisms. Particularly, we show that quasihomeomorphisms preserve Whyburn, weakly Whyburn, submaximal and door properties. Moreover, we offer necessary conditions on continuous map q : X → Y where Y is Whyburn (resp., weakly Whyburn ) in order to render X Whyburn (resp., weakly Whyburn). Also, we prove that if q : X → Y is a one-to-one continuous map and Y is submaximal (resp., door), then X is submaximal (resp., door). Finally, we close this paper by studying the relation between quasihomeomorphisms and k -primal spaces.


Introduction and Preliminaries
The concept of quasihomeomorphism was first introduced by the Grothendieck school [1,2].This notion is mainly used in algebraic geometry, and it was shown that this concept arises naturally in the theory of some foliations associated to closed connected manifolds [1].
We recall that subset L of topological space X is called locally closed if it is an intersection of an open set and a closed set of X.We take O(X), F (X), and L(X) as the families of all open, closed, and locally closed subsets of X, respectively; we call a continuous map q : X → Y a quasihomeomorphism if A → q −1 (A) represents a bijection from O(Y) (resp., F (Y), L(Y)) to O(X) (resp., F (X), L(X)).
Topological space X is called Whyburn [3] if for every non-closed subset A of X and for every x ∈ A\A there exists B ⊆ A such that B\A = {x} or, equivalently, there exists B ⊆ A such that B\B = {x}.It is called weakly Whyburn [4] if for every non-closed subset A of X there exists B ⊆ A such that | B\A |= 1 or, equivalently, there exists B ⊆ A such that | B\B |= 1.It was illustrated that Whyburn space is weakly Whyburn, whereas the converse side is not always true; see [5], Theorem 3.8.
A door space is a topological space in which every subset is either open or closed.By a submaximal space, we mean a topological space in which every subset is locally closed or, equivalently, every dense subset is open.
A principal space, which is also recognized as Alexandroff space, is a topological space in which any intersection of open sets is open.The most fundamental property of Alexandroff spaces is that the category of Alexandroff spaces is isomorphic to the category of qosets.We let Alx denote the category of Alexandroff spaces and Qos the category of quasi-ordered sets.We let φ : Qos −→ Alx be the map defined by φ((X, ≤)) = (X, τ ≤ ) where τ ≤ is the specialization topology defined by choosing, B = {(x ↑) x ∈ X} as a basis of τ ≤ , where (x ↑) = {y ∈ X; x ≤ y} is called the upset determined by x.In this context, the closure {x} τ ≤ is exactly the downset (↓ x) = {y ∈ X : y ≤ x}.Therefore, it is clear that φ((X, ≤)) is an Alexandroff space.Similarly, we define ψ : Alx −→ Qos by ψ((X, τ)) = (X, ≤ τ ) where ≤ τ , called corresponding specialization quasi-order, is defined by for any x, y ∈ X; x ≤ τ y if and only if x ∈ {y} τ .Clearly, φ and ψ are inverse maps one of the other, which means that considering an Alexandroff space is equivalent to considering a qoset.
In this paper, we detect topological properties preserved by quasihomeomorphisms.We show that the Whyburn (resp., weakly Whyburn, Submaximal, door) property is preserved by quasihomeomorphisms.Furthermore, we lay bare the necessary conditions on continuous map q : X → Y where Y is Whyburn (resp., weakly Whyburn) to ensure that X is Whyburn (resp., weakly Whyburn).
This paper is organized as follows.In Section 2, we show that quasihomeomorphisms preserve the properties of being Whyburn, weakly Whyburn, submaximal, and door spaces.We devote Section 3 to revealing the necessary conditions on continuous map q : X → Y where Y is Whyburn (resp., weakly Whyburn, submaximal, door) in order to assure that X is Whyburn (resp., weakly Whyburn, submaximal, door).Finally, in Section 4, we prove that given a quasihomeomorphism, q : X → Y between two Alexandroff spaces.Then, if q is an onto (resp., one-to-one) and X (resp., Y) is strongly primal, so is Y (resp., X).

Quasihomeomorphisms and Some Topological Properties
In this section, we detect topological properties preserved by quasihomeomorphisms.Firstly, let us start by recalling the following results.Lemma 1 ([6]).We let q : X −→ Y be a continuous onto map.Then, the following statements are equivalent: 1. q is a quasihomeomorphism; 2. q is open and equality q −1 (q(U)) = U holds for any open set U in X; 3. q is closed and equality q −1 (q(F)) = F holds for any closed set F in X.

Lemma 2 ([7]
).If q : X −→ Y is a quasihomeomorphism, then the following statements are equivalent: 1. q is onto; 2. q −1 (A) = q −1 (A) for any set A in Y. Theorem 1.We let q : X −→ Y be an onto quasihomeomorphism.If X is Whyburn, then Y is Whyburn.
Proof.We let A be a non-closed subset of Y and y ∈ A\A.Since q is an onto, there exists x ∈ X such that y = q(x) ∈ A\A.It can be seen easily that x ∈ q −1 (A)\q −1 (A).Using the fact that X is Whyburn, we let B be a subset of q −1 (A) that satisfies B\B = {x}.Let us show that q(B)\q(B) = {y}.Since q is continuous and closed, we obtain q(B) = q(B) and thus y = q(x) ∈ q(B).Now, we suppose that y = q(x) ∈ q(B); then, x ∈ q −1 (q(B)) ⊆ q −1 (q(q −1 (A))) = q −1 (A) is a contradiction, so that y ∈ q(B)\q(B).
In the following example, we show that the surjectivity of quasihomeomorphism q is necessary to conclude that Y is Whyburn.
The following example shows that if in Theorem 1, q is not a quasihomeomorphism, then Y need not be a Whyburn space.
We let Y = {a, b, c, d} be a space whose basis of open sets is {{c}, {d}, {b, d}, {a, b, d}}.
Then, X is a Whyburn space and Y is not Whyburn.We let q : X → Y be defined by q(0) = q(2) = d, q(1) = a, q(3) = b and q(4) = c.Hence, q is an onto continuous map and q −1 (O) = {3} whenever O is an open set in Y. Thus, q is not an quasihomeomorphism.
Next, we show that the property weakly Whyburn is preserved by onto quasihomeomorphisms.
Theorem 2. We let q : X → Y be an onto quasihomeomorphism.If X is weakly Whyburn, then Y is weakly Whyburn.
Proof.We let A be a non-closed subset of Y. Since q is an onto quasihomeomorphism, then q −1 (A) is a non-closed subset of X. Regarding X being weakly Whyburn, we consider set Therefore, Y is weakly Whyburn.

Remark 1. (1)
The fact that q is a quasihimeomorphism in Theorem 2 is necessary as shown by Example 2. Of course, X is weakly Whyburn.However, Y is not.Indeed, | {d}\{d} | = 1 (2) The condition of suyjectivity in Theorem 2 is necessary as shown by Example 1. Theorem 3. We let q : X → Y be an onto quasihomeomorphism.If X is submaximal, then Y is submaximal.
Example 3. We let X = {0, 1} be equipped with topology {∅, X, {0}}.Then, the dense subset {0} is open in X and thus X is submaximal.Now, we let Y = {0, 1, 2} be a finite space whose open sets are ∅, Y and {0}.Then, we have {0, 1} as a dense subset of Y and it is not open.We let q : X → Y be the canonical injection.It can be seen easily that q is a quasihomeomorphism that is not an onto.Example 4. We let X = {0, 1, 2} be a space whose open sets are ∅, X, {1, 2}, {2} and {1}.Then, every dense subset of X is open and thus X is submaximal.We let Y = {0, 1, 2} be equipped with topology {∅, Y, {0}}.It can easily be seen that Y is not a submaximal space.Now, we let q : X → Y be the identity map.Thus, q is an onto that is not quasihomeomorphism.Theorem 4. We let q : X → Y be an onto quasihomeomorphism.If X is door, then Y is door.
Proof.We let A be a subset of Y. Since X is door, we obtain q −1 (A) that is either open or closed.Hence A = q(q −1 (A)) is either open or closed in Y.
Remark 2. We remark that in Theorem 4, it is enough to suppose that q is an onto closed continuous map.

Inverse Preserved Topological Properties
We let q : X → Y be a continuous map.In this section, we are interested in additional conditions to X in order to satisfy properties satisfied by Y. Hence, we introduce the following definition: Definition 1. Continuous map q : X → Y is said to be satisfying the complement closed property if and only if q −1 (A) is not closed in X whenever A is not closed in Y.
We recall that Alexandroff spaces are topological spaces in which arbitrary intersections of open sets are open.It is clear that if space X is Alexandroff, then A = [{x} : x ∈ A], for every subset A of X. Theorem 5. We let q : X → Y be a one-to-one continuous map from an arbitrary topological space X to a Whyburn space Y.Then, the following properties hold: (1) If X is Alexandrov, then it is Whyburn.
(2) If q satisfies the complement closed property, then it is Whyburn.
Example 7. We let X = {1, 2, 3} and Y = {α, β} both be equipped with indiscrete topology.Then, X is not Whyburn and Y is a Whyburn space.We let q : X → Y be defined by q(1) = q(3) = α and q(2) = β.It may be checked easily that q is continuous and q is not one-to-one.Now, we study the weakly Whyburn property.Theorem 6.We let q : X → Y be a one-to-one continuous map from an arbitrary topological space X to a weakly Whyburn space Y.Then, the following properties hold: (1) If X is Alexandrov, then it is weakly Whyburn.
(2) If q satisfies the complement closed property, then it is weakly Whyburn.
Theorem 7. We let q : X → Y be a one-to-one continuous map.Then, the following properties hold: (1) If Y is submaximal, then X is submaximal.
(2) If Y is door, then X is door.

Proof. (1)
We let A be a dense subset of X.Then, q(A) is a dense subset of q(X).But it can easily be seen that q(A) ∪ (Y\q(X)) is a dense subset of Y and since Y is submaximal, We let A be a subset of X; then, A = q −1 (q(A)), which is open or closed.Example 8. We let X = {0, 1, 2} be a space whose open sets are ∅, X and {0}.We let Y = {0, 1} equipped with topology {∅, {0, 1}, {0}}.Subset {0, 2} is a dense subset of X and it is not open, and thus X is non-submaximal.On the other hand, it can be seen easily that Y is a submaximal space.Now, we let q : X → Y be defined by q(0) = 0, q(1) = q(2) = 1.Then, q is a continuous map and q is non-one-to-one.

Quasihomeomorphisms and k-Primal Spaces
Ayatollah Zadeh Shirazi and Golestani [8] on the one hand and Echi [9] on the other hand, working independently, explicitly introduced a class of Alexandroff topologies on X called by Echi primal spaces and by Shirazi and Golestani functional Alexandroff spaces.In this paper, we use the terminology of primal spaces.Given map f : X −→ X, we define the Alexandroff topology on X by taking the closure of point x ∈ X in this topology orbit { f n (x) : n ∈ N}, where N is the set of all natural numbers including 0. Therefore, subset A in this topology is closed if and only if it is an f -invariant set, that is, f (A) ⊆ A. Equivalently, in the language of qosets, we define (X, ≤ f ) by for any x, y ∈ X, y ≤ f x if and only if y = f n (x) for n ∈ N.Such a topology is denoted by (X, P ( f )) = (X, ≤ f ).
The equivalence between Alexandroff spaces and quasi-ordered sets is the motivation to introduce the following definition.
In [9], Echi introduced the notion of primal spaces as follows: Definition 2. We let X be a non-empty set and f be a map from X to itself.Subset A of X is said f -invariant if f (A) ⊂ A. The family of all f -invariant sets forms closed sets of a topology on X called primal topology and denoted by (X, P( f )).
In [11], the authors showed that when giving a quasihomeomorphism q from X to Y, if q is onto and X is primal, then Y is primal, and if q is one-to-one and Y is primal, then so is X.Some examples showing the importance of conditions of quasihomeomorphism, onto and one-to-one, were given (for more information, see [11,18]).
On the other hand, in [19], the authors introduced the notion of k-primal spaces as follows: Definition 3.For any non-zero positive integer k and any family { f i : 1 ≤ i ≤ k} of functions from a given set X to itself, we define the k-primal space (X, P ( f 1 , . . ., f k )) = (X, ≤ f 1 ,..., f k ) as the intersection of primal spaces (X, P ( f i )) = (X, ≤ f i ); 1 ≤ i ≤ k.That is, for any x, y ∈ X, x ≤ f 1 ,..., f k y if and only if x ≤ f i y for every 1 ≤ i ≤ k.
Hence, the following definition is immediate: Definition 4.An Alexandroff space is called a strongly primal space if and only if it is a k-primal space for some positive integer k.
The characterization of weakly primal spaces, in finite cases, is given by [19].Theorem 8. We let (X, ≤) be a finite qoset.Then, (X, ≤) is a k-primal space for some k ∈ N, k = 0 if and only if for every cyclic point a and every x, y ∈ X we have a ≤ y x ≤ y =⇒ a ≤ x.
In this section, we study the relation between strongly primal spaces and quasihomeomorphisms in the class of finite Alexandroff spaces.
Before offering the main result of this section, we need the following lemma: Lemma 3. We let q : X −→ Y be quasihomeomorphisms between two Alexandroff spaces.Then, for every x, y ∈ X, x ≤ y ⇐⇒ q(x) ≤ q(y).
Proof.If x ≤ y, then x ∈ {y}, which is included in {q(y)} by continuity of q and thus q(x) ≤ q(y).Conversely, we let x, y ∈ X such that q(x) ≤ q(y).Since q is a quasihomeomorphism, there exists a unique closed subset F of Y satisfying {y} = q −1 (F).Now, using q(y) ∈ F and q(x) ≤ q(y), we obtain q(x) ∈ F and thus x ∈ {y} = q −1 (F) which means that x ≤ y.Now, we are in a position to cite and prove our main theorem in this section.Theorem 9. We let q : X −→ Y be quasihomeomorphisms between two finite Alexandroff spaces.Then, the following properties hold: (1) If q is onto and X is strongly primal, then so is Y.
(2) If q is one-to-one and Y is strongly primal, then so is X.
Proof.(1) We let b be a cyclic point in Y, x and y two points such that b ≤ y and x ≤ y .
Since q is an onto, there exist a, x, y ∈ X with b = q(a), x = q(x) and y = q(y).It can easily be seen that a is a cyclic point in X such that a ≤ y and x ≤ y.Now, since X is strongly primal, a ≤ x and consequently, b = q(a) ≤ q(x) which complete the proof.(2) Now, we consider cyclic point a in X and x, y ∈ X with a ≤ y and x ≤ y.Since q is one-to-one and a is cyclic, then q(a) is a cyclic point in Y such that q(a) ≤ q(y) and q(x) ≤ q(y).Hence, by the fact that Y is strongly primal, we obtain q(a) ≤ q(x).Therefore, a ≤ x and X is strongly primal.

Example 9.
(1) We consider X = {0, 1} equipped with the indiscrete topology and Y is an infinite set equipped with the indiscrete topology.We let q from X to a Y be a constant map.Clearly, q is a non-onto quasihomeomorphism.However, X is primal and thus a strongly primal space but not Y (see [19], Proposition 2.2).This example shows that the surjectivity of q in Theorem 9 is necessary.(2) We consider X an infinite set equipped with the indiscrete topology and Y = {0, 1} equipped with the indiscrete topology.We let q from X to a Y be a map which associates zero to a fixed point x ∈ X and one to the complement of {x}.Clearly, q is a non-one-to-one quasihomeomorphism.However, Y is primal and thus a strongly primal space, but not X.This example shows that the injectivity of q in Theorem 9 is necessary.

Conclusions
In this paper, we investigated some topological properties that are preserved by surjective quasihomeomorphisms such as Whyburn, weakly Whyburn, submaximal and door spaces.On the other hand, we demonstrated the necessary conditions that guarantee to keep these properties under the pre-image continuous.In the end, we discussed the relationships between quasihomeomorphisms and k-primal spaces.
As it is well known that topology has been exploited to address many real-life problems via different fields [20][21][22], we hope the results obtained herein open the way to conduct further studies on both sides theoretical and application.It will be interesting to investigate the current concepts in the frame of soft settings and examine their properties and characterizations.