1. 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 [3–6]).
The specialization quasiorder of an Alexandroff space is defined by
Now, if is a quasiorder on space then the set of all supersets (.) forms a basis of an Alexandroff topology on . In this case, the closure is exactly the downset . We denote by the minimal neighborhood of . For more information on Alexandroff spaces you can see [7–13].
In [14], Echi introduced a particular class of Alexandroff spaces named primal spaces. is called a primal space if there exists a map such that , where is the collection of all -invariant subsets of (for more information see [14, 15]). In [16], the authors characterized maps such that the primal space is submaximal or door.
This paper is devoted to characterizing Alexandroff spaces which are submaximal, door, -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 -resolvable. The second section is devoted to introducing and characterizing topological spaces, called quasi-Whyburn spaces, such that their -refections are Whyburn. Particularly, the case of Alexandroff spaces is totally deduced in this particular class of spaces.
2. Submaximal, Door, and -Resolvable Alexandroff Spaces
We know that a submaximal space is a topological space in which all dense subsets are open.
The following theorem characterizes submaximal spaces [17].
Theorem 1. A topological space is submaximal if and only if for every , the subset is closed equivalently for every and the subset is closed and discrete.
Now, before giving the first main result of this section, some useful examples are presented and commented.
Example 1. (1)Let , and be distinct points and equipped with the topology . We can find the specialization quasiorder as follows: , , , and so that , , , and . We have is an Alexandroff space which is submaximal.(2)Consider the set of natural numbers and for each , let . We equip the space with the topology .The Alexandroff space is not submaximal. In fact, is a dense subset which is not open.
The previous examples give a motivation to investigate conditions allowing Alexandroff spaces to be submaximal.
The answer is given by the following theorem.
Theorem 2. Let be an Alexandroff space. Then, the following statements are equivalent:(1) is submaximal(2)The specialization quasiorder is an order and every chain in its graph has a length less than or equal to 2
Proof. (i). Suppose that is submaximal. If are two elements of satisfying and then and , which imply that . Using the fact that every submaximal space is , we deduce that and then the specialization quasiorder is an order. Now, suppose that . If , then and . Since then it is not closed.(ii). If is a subset of , we denote . Using 2), for every and for any , is closed. Since X is an Alexandroff space, then so that is closed and then is submaximal.
Corollary 1. (see [14], Theorem 4.1)
If is a flow in set, then is a submaximal space if and only if .
Definition 1. Let be a topological space. is called a door space if any subset of is open or closed.
Now, we state straightforward remarks.
Remark 1. (1)The example cited in Example 1 (1)provides a space that is not a door. Indeed, the subset is neither open nor closed.(2)Let be the set equipped with the topology defined as follows: for each and .Hence, every subset of not containing 0 is open. Yet, every subset of containing 0 is closed.
Therefore, 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.
The increasing hull of a set in a quasiordered set is . A set is increasing if . The set may be written as . Decreasing hulls and decreasing sets are defined dually. The closed sets in an Alexandroff space are just the decreasing sets for the specialization quasiorder, and the open sets are just the increasing sets.
Theorem 3. Let be an Alexandroff space and is its specialization order. Then, the following statements are equivalent:(1) is a door space(2)The length of every chain in the graph of is not greater than 2 and all chains of length 2 contain a common point which must be a maximal point or a minimal point
Proof. (i). Suppose that the common point is a minimal point . Let be a nonclosed subset of . So that is not decreasing in equivalently there is such that and . Since a subset which not contains is increasing, therefore, it is open. We work dually if the common point is a maximal point.(ii). By contradiction suppose that either there exist or and of length 2 with no common point. In these cases, we have always is not increasing and not decreasing, thus it is not open and not closed which is a contradiction. This fact completes the proof.
Corollary 2. (see [16], Theorem 4.3)
If is a flow in set, then is a door space if and only if .
Now, we will give a study of Alexandroff spaces.
First, let us recall the definition of spaces. If is a topological space, then it is called if there exist -many mutually disjoint dense sets of . 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 if and only if it is the union of -many mutually disjoint dense subsets.
Stone [18] characterizes Alexandroff spaces which are -resolvable in the following theorem.
Theorem 4. (see [18]). Let be a quasiordered set. Then, we have equivalence between the following statements:(1) admit a partition into n mutually disjoint cofinal sets(2), has at least elementsWe note that, for every subset of an Alexandroff space , we have equivalence between the following items:(1) is dense in (2)(3), such that (4) is cofinal in This allows us to rephrase Stone’s result as follows.
Theorem 5. Let be an Alexandroff space and is its specialization quasiorder. Then, the following statements are equivalent:(1)“” is -resolvable(2), contains at least elements(3)There is no maximal element in (4) has no isolated pointsWe recall that the -reflection of a topological space is the quotient space denoted by obtained from the equivalence relation defined on by if and only if .
Corollary 3. Let be an Alexandroff space. Then the following statements are equivalent:(1) is -resolvable(2), contains at least distinct points(3)Every maximal element in the -refection arises from a cycle containing at least distinct points Now, we shed some light on interesting examples.
Example 2. (1)Consider the set of all integers with the usual order . For any integer let . Then are mutually disjoint dense sets of the Alexandroff space , showing that this space is -resolvable for every . Indeed, it is obvious that is infinite for each .(2)Let be the inverse order of where is the set of all natural numbers. In the Alexandroff space , every set of is dense if and only if . Therefore, is not a resolvable space. In fact, we note that .
3. Alexandroff Spaces Which Are Whyburn and Quasi-Whyburn Spaces
Let be a topological space and be a subset of . Then, is called almost closed if and only if for some . We use the notation .
The 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 satisfying
A topological space is called weakly Whyburn [20] if
We denote the class of all Whyburn spaces (resp., weakly Whyburn spaces) by AP-spaces (resp., WAP-spaces) [21–23].
3.1. Quasi-Whyburn Spaces
A continuous map from a topological space to a topological space is said to be a quasihomeomorphism if defines a bijection between the collection of all open sets of and the collection of all open sets of [24].
We can see easily that the canonical surjection is a quasihomeomorphism. More precisely, is an onto quasihomeomorphism, and in this case, the following results are useful.
Lemma 1. (see [25]). Let be continuous onto the map. Then, is a quasihomeomorphism if and only if is an open map and for every open subset of ; equivalently, is a closed map and for every closed subset of .
Lemma 2. (see [16]). A quasihomeomorphism is onto if and only if for every subset of .
If is a topological space, , and , we take the notations in [16]. In that paper, authors denote by the subset and by the union of for all .
Using these notations, we can find the following properties:(i)(ii)(iii) and (iv)If is open or closed, then
Now, we introduce the notions of -closed subsets, in a given topological space, and quasi-Whyburn spaces as follows.
Definition 2. Let be a subset of a topological space . Then,(i) is called -closed if is closed, that is, if (ii)If the -reflection of is a Whyburn space, is called a quasi-Whyburn space or a -space (or also a -Whyburn space)The following theorem gives a characterization of quasi-Whyburn spaces.
Theorem 6. If is a topological space, then we have equivalence between the following statements:(1) is a -space(2)For all non--closed subsets of and for all , there is a subset of such that with
Proof. (1) (2). Let be a non--closed subset of and . Then, . By hypothesis, there is such that (which is equivalent to ) satisfying . Now, applying , we have(2) (1). Conversely, let such that is not closed in and consider a point in with . Then, with non--closed. So, by hypothesis, there is a subset of such that (which is equivalent to ) satisfying . Thus, . Therefore, .
Definition 3. A topological space is called quasiweakly Whyburn space (or -weakly Whyburn space) and denoted by -space if its -reflection is a weakly Whyburn space.
The proof of the following result is similar to that of Theorem 6.
Theorem 7. If is a topological space, then we have equivalence between the following statements:(1) is a -space(2)For all non--closed subset of , there is a subset of with and , for some
3.2. Alexandroff Spaces Which Are Whyburn and Quasi-Whyburn Spaces
Theorem 8. If is an Alexandroff space, then we have equivalence between the following statements:(i) is Whyburn(ii)
Proof. Suppose that is Whyburn and there exists such that contains two distinct elements and . Since is not closed and , there exists such that . Yet, in that case, , which leads to a contradiction because contains also .
Conversely, suppose that each element of has at most 2 predecessors.
Let such that . Using the fact that is Alexandroff, we haveLet and satisfying . Since , then . If we take , we can see that and . We deduce that is a Whyburn space. □
Corollary 4. Let be a functionally Alexandroff space. Then, we have equivalence between the following statements:(i) is a Whyburn space(ii)
Example 3. (1)Consider a given set . Let be an Alexandroff topology on . Suppose that is Whyburn. Since is not closed, then there exists such that . This is an impossible fact because . Therefore, this Alexandroff space is not Whyburn.(2)Let . The topology on satisfying , for every and is an Alexandroff topology.Clearly, for any nonempty subset of , we have ; then, the condition means that , and thus, . We observe that, in this case, for every , and . One can illustrate this situation in Figure 1.
Remark 2. (1)Let be the Alexandroff space cited in Example 1 (1) and suppose that it is Whyburn. Since is not closed, then there exists such that . This is an impossible fact because . Therefore, this Alexandroff space is not Whyburn. Moreover, for the same reason, we note that the space cited in Example 1 (2) is not Whyburn.(2)The example cites in Remark 1 (2) provides a Whyburn space.
Proposition 1. Let be an Alexandroff space. Then, is weakly Whyburn if and only if is Whyburn.
Proof. Clearly, any Whyburn space is weakly Whyburn.
Conversely, suppose that is a WAP-space and there exists such that .
If we take , then is not closed, which implies that there exists such that . Thus, , and so which is a contradiction.
Corollary 5. Let be a Whyburn space. Then, the following statements are equivalent:(1) is an Alexandroff space(2) is a primal space
Proof. It is enough to see that an Alexandroff Whyburn space is a functionally Alexandroff space. Hence, by Theorem 8, for any . Two cases arise as follows:(a)If , we take , and thus .(b)If , then , and in this case, we take . So, = . Indeed, since which is equivalent, by the construction of .(i), if , and thus (ii), if , and thus , and consequently,
Theorem 9. Let be an Alexandroff space. Then, the following statements are equivalent:(i) is a -space(ii) and , we have
Proof. The first remark that is Alexandroff if and only if is Alexandroff and for any ; we have .(i). Let and . Then, . Now, using Theorem 8, we get . Therefore, .(ii). Let . By Theorem 8, it is enough to see that . In this case, suppose that . Then, , and thus, by hypothesis, the family is not pairwise distinct, as desired.
Example 4. (1)Figure 2shows the following: and . Let , then two cases arise as follows:(i)If , then necessary or (ii)If , then means that and thus (2) Figure 4 shows the following:, , and . Then, , but is a family of pairwise distinct elements. Therefore, is not a -space. is defined by Figure 5which is not a functionally Alexandroff space.(3)Figure 6 shows the following: , , and thus, for any , we have . Therefore, is a -space. is defined by Figure 7 which is a functionally Alexandroff space.
Theorem 10. Let be an Alexandroff space. Then, the following statements are equivalent:(i) is a -space(ii) is a -space
Proof. Using Proposition 1 and the fact that a topological space is Alexandroff if and only if its -reflection is Alexandroff. We get it immediately.
Data Availability
The data set can be accessed upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This project was supported by the Deanship of Scientific Research at Prince Sattam bin Abdulaziz University under the research project 2020/01/16 517.