Epi--Normality and Epi--Normality

Abstract

A topological space is called epi--normal (epi--normal) if there is a coarser topology on such that is -normal (-normal). We investigate these properties and show some examples to explain the relationships of epi--normal (epi--normal) with other weaker versions of normality and some topological spaces.

1. Introduction

In [1], the notions of -normality and -normality were investigated in relative sense, and their dependency and independency on other generalizations of relative normality were proved. Some relative generalizations of -normality are defined, their relations with different generalizations of relative normality were studied, and it is observed that some absolute generalizations of normality coincide with normality in the class of relative variant of -normal. Also, almost -normal spaces as a simultaneous generalization of -normal and almost normal spaces were introduced by Das et al. [2]. Another new generalization of normality, namely, weak -normality, in terms of -closed sets was introduced, which turns out to be a simultaneous generalization of -normality and -normality (see [3]).

In this paper, we introduce new topological properties called epi--normality and epi--normality. We investigate these properties and show some examples to explain the relationships between epi--normality (epi--normality) with other weaker versions of normality and some topological spaces. We prove that submetrizablity, imply epi--normality and -normality implies epi--normality, and we prove the converse is not true in general. Throughout this article, a space is a -normal space, a completely regular space with is Tychonoff space, and regular space with is .

2. Epi--Normality and Epi--Normality

Recall that a topological space is called epinormal [4] if there is a coarser topology on such that is . A topological space is called -normal [5] if for any two disjoint closed subsets and there exist disjoint open subsets and of such that is dense in and is dense in . A topological space is called -normal [5] if for any two disjoint closed subsets and there exist open subsets and of such that is dense in , is dense in , and .

Definition 1. A topological space is called epi--normal (epi--normal) if there is a coarser topology on such that is -normal (-normal).
Note, if we assume to be just -normal (-normal) with out assume in the above definition, therefore any space will be epi--normal (epi--normal) as the indiscret topology.
Obviously, any -normal space is epi--normal, any -normal space is epi--normal, and any space is epi--normal just by taking , and any epinormal space is epi--normal since every normal space is -normal.
Note that any submetrizable space [4] is epi--normal, but the converse is not true in general. For example, is an epi--normal space is Hausdorff and compact; therefore, . But it is not submetrizable (for details, see [4]).
Epi--normality and -normality do not imply each other. For example, the Niemytzki plane is epi--normal since it is submetrizable, but it is not -normal by the standard proof of nonnormality. On the other hand, an example of an -normal space which is not epi--normal is any indiscrete space which has more than one point.
Observe that if the coarser topology in Definition 1 is , , then so is . Since any -normal (epi--normal) is Hausdorff [5], we conclude the following.

Theorem 1. Any epi--normal (epi--normal) space is Hausdorff.
Note that every topological space which is not Hausdorff can neither be epi--normal nor be epi--normal. For example, the Double pointed reals topological space defined in [6] is -normal, and hence normal but neither epi--normal nor epi--normal since it is not Hausdorff.

Theorem 2. Every epi--normal space is completely Hausdorff.

Proof. Let be an epi--normal space. Let be a coarser topology on ; then, is -normal. We assume that has more than one element and take distinct . Since is -normal, then it is (see [5]). Choose such that , , and . Now since is -normal, then by Proposition 1.1 in [5], is regular, and therefore choose such that and . We have and since for any , we get .
Note that any space which is not completely Hausdorff cannot be epi--normal. For example, Prime Integer topology [6]. In this space, the set of positive integers, where any open neighborhood , since the closures of any two open neighborhood , contain in common all multiples of , so it is not completely Hausdorff (see [6]), and by Theorem 2, the space cannot be epi--normal.
Observe that epi--normality may not be necessarily completely Hausdorff. However, we have the following theorem.

Theorem 3. If is an epi--normal space and the coarser topology of epi--normality ) is first countable, then is completely Hausdorff.

Proof. By the similar argument of Theorem 2, since every Hausdorff -normal first countable is regular, the proof follows (see [7]).
Since every -normal space satisfying axiom is regular (see [5]), we recall that a topological space is called epiregular [8] if there is a coarser topology on such that is , so easily we conclude the following.

Corollary 1. Any epi--normal space is epiregular.
Any paracompact space is , so we conclude the following.

Corollary 2. If is an epi--normal space and the coarser topology of epi--normality is paracompact, then is .
Also, it is well known that any compact space is , so we conclude the following.

Corollary 3. Any epi--normal compact space is .
Recall that a topological space is called -regular [9, 10] if for any closed subsets of and such that there exist disjoint open subsets and of such that and is dense in . A topological space is called epi--regular (see [9]) if there is a coarser topology on such that is -regular. Since every -normal space satisfying axiom is -regular (see [9, 10]), we conclude the following.

Corollary 4. Every epi--normal space is epi--regular.

Theorem 4. If is an epi--normal pseudocompact space of cardinality less than the continuum and the coarser topology of epi--normality is completely regular, then is .

Proof. Let be an epi--normal pseudocompact space of cardinality less than the continuum and be a coarser topology on where is -normal. Note that is pseudocompact. Since any -normal pseudocompact space is countably compact (see Theorem 2.2 in [5]), is countably compact Hausdorff. Since it is of cardinality less than the continuum, then is first countable.
Now, we show that . Let be any closed set in . Assume that the family . Clearly we have . Since the pseudocompactness is hereditary with respect to closed domain ([11], 3.10. F), is pseudocompact in for every . Then, is pseudocompact in for every . Since pseudocompact subsets of first countable Tychonoff space are closed [12], is closed in for every . Hence, is closed in and then . Thus, is .

Corollary 5. If is an epi--normal pseudocompact first countable space and the coarser topology of epi--normality is completely regular, then is .

Theorem 5. If is an epi--normal space and the coarser topology of epi--normality is a dense subspace of a product of metrizable spaces, then is epinormal.

Proof. Assume that is an epi--normal space; then, there is a coarser topology such that is -normal and it is a dense subspace of product metrizable spaces. Śćepin and Blair independently showed that every dense subspace of any product of metrizable spaces is mildly normal [5, 13, 14]. Since is -normal and mildly normal, then by Theorem 2.4 in [5], is normal and hence . Therefore, is epinormal.
The following is an example of normal Hausdorff non-regular space constructed by Murtinová in [7] where she showed that it is an example of an normal Hausdorff non-regular space.

Example 1 (see [7]). Let and define a topology such that with the ordinal topology is an open subspace and a base in the point will be the collection:where is a closed unbounded subset of (Club).
The topology is Hausdorff since it is stronger than the order topology on . This space is epi--normal since it is -normal Hausdorff and it is epi--normal since it is stronger than the order topology on , but it is not -normal nor regular.
Murtinová in [7] proved that every first countable -normal Hausdorff space is regular, so we have the following result.

Theorem 6. If is epi--normal and the coarser topology of epi--normality is first countable, then is epiregular.
Recall that a topological space is called epicompletely regular [15] if there is a coarser topology on such that is Tychonoff.

Theorem 7. If a topological space is epicompletely regular and the coarser topology of epicompletely regularity is second countable (countable), then the space is epi--normal.

Theorem 8. If a topological space is epi--normal and the coarser topology of epi--normal is compact second countable, then the space is submetrizable.
The similar argument is correct for epi--normality.

3. Dowker’s Characterisation of Epi--Normal and Epi--Normal

In light of Dowker’s characterisation of -normal and -normal (for details, see [16]), it is natural to ask if this can be satisfied for epi--normal and epi--normal. To do this, let us start with the following lemma from [16]. Throughout this section, unless otherwise stated, a space is assumed to be an epiregular topological space.

Lemma 1 (see [16]). A topological space is -normal if and only if for every pair and of disjoint closed subsets of there exists an open set of such that and is nowhere dense in .

Proof. Let be an -normal space and , be disjoint closed subsets of . Then, there are two disjoint open subsets and such that and . It follows that which means that . Now let ; then, there is at least an element , so , which means that and ; then, there is an open set containing and contained in where . This contradicts . Therefore, . Hence, is nowhere dense in .
Conversely, let and be disjoint closed subsets of ; there exists an open set of such that and is nowhere dense in . Then, which implies that . Now let ; then, . It follows that for any open subset in containing , is not contained in and then . Let . Then, is an open set disjoint from and , and so . Hence, is an -normal space.
The following theorem has the same argument of Theorem 2.3 in [16].

Theorem 9. Let be the order topology defined on . The product is epi--normal if and only if(1) is epi--normal.(2)Let be the coarser topology of epi--normality. If is a family of closed sets in and and is a closed subset of disjoint from , then there is a family of open sets in such that is dense in and is nowhere dense in .The case of epi--normal spaces is similar to the case of epi--normal spaces, but it is more complicated. First, we introduce the following lemma induced from [16].

Lemma 2. Let be a space. If is epi--normal, then(1) is -normal.(2)Let be the coarser topology of epi--normality. If is a family of closed sets in and and is a closed subset of disjoint from , then there is a family of open sets in such that is dense in and is disjoint from .Recall that a topological space is said to be -countably paracompact [16] if for every decreasing sequence of closed subsets of satisfying there exists a sequence of open subsets of such that is dense in for and .
With the definition of -countably paracompact from [16] and Lemma 2, it is not hard to prove the following corollary.

Corollary 6. If is epi--normal, then Y is epi--normal and -countably paracompact.
With Lemma 2, we are ready to introduce an important result of epi--normal spaces which has the same argument of [16].

Theorem 10. Let be the order topology defined on . The product is epi--normal if and only if(1) is epi--normal.(2)Condition of Lemma 2 is satisfied.(3)Let be the coarser topology of epi--normality. For each decreasing sequence of closed sets in satisfying , there is a sequence of open subset of where for and is nowhere dense in the relative topology of with respect to the coarser topology .

Corollary 7. Let be epi--normal and countably paracompact. Then, is epi--normal.

Corollary 8. Let be epi--normal and countably paracompact. Then, is epi--normal.

4. Properties of Epi--Normality and Epi--Normality

Let be any topological space. Let . Note that . Let . For an element , denoted by , and for a subset , we have . For each , let . For each , let . We define a basic neighborhood for a unique topology defined on . This generated topological space is called Alexandroff duplicate of (for revision, see [17]).

Theorem 11. Let be an epi--normal space, and so is its Alexandroff duplicate .

Proof. Let be any epi--normal space and let be a topology on ; since is epi--normal, then there is a coarser topology on such that is -normal. Since by Theorem 0.1 in [18] -normality is preserved by the Alexandroff duplicate space and also , then is also -normal, and it is coarser than by the topology of the Alexandroff duplicate. Hence, is epi--normal.
We do not know if the above theorem is valid for epi--normal space.

Theorem 12. -normality and -normality are additive properties.

Proof. We prove the case for -normality, and that for -normality will be similar.
Let be a family of -normal spaces and , be two disjoint closed subsets of the sum . By Proposition 2.2.1 in [11], the intersections and are closed in for each . From -normality of , it follows that there are two open sets and in whereLet and ; then, clearlySince and are open in , the sum is -normal.

Theorem 13. Epi--normality and epi--normal space are additive properties.

Proof. Let be epi--normal spaces for each . Let be a topology on coarser than such that is -normal. Since and -normality is also additive by Theorem 12. Then, and the sum is and -normal, and its topology is coarser than the topology on . The case for epi--normality is similar.

Theorem 14. Let be a -normal (-normal) space, and is an onto, continuous, open, and closed function. Then, is -normal (-normal).

Proof. We prove the case for -normality, and that for -normality will be similar. Let and be disjoint closed subsets of . Then, by continuity of , and are disjoint closed subsets of . Since is a -normal space [5], there exists an open subset of such that and . Since , then . Observe that is a closed set containing the open set and . Then, . Now it is sufficient to show that . Let and be an open set containing ; then, . Since , . Hence, by surjectivity of , as required.

Corollary 9. Let be an epi--normal (epi--normal) space, and is an onto, continuous, open, and closed function. Then, is epi--normal (epi--normal).

Proof. We only prove the case for epi--normality. Let be any epi--normal space, and let be a coarser topology on such that is -normal, . Since is an onto, continuous, open, and closed function, then by Theorem 14, where , is -normal, and it is obviously and is coarser than . Hence, is epi--normal.

Corollary 10. Epi--normality and epi--normality are topological properties.

Theorem 15. Let be an epi--normal (epi--normal) space, and is a closed subspace of the coarser topology of epi--normality (epi--normal) ; then, the subspace is epi--normal (epi--normal).

Proof. This is straightforward since any closed subspace of an -normal (-normal) space is -normal (-normal) (see Proposition 1.3 in [5]).

Definition 2. A space is hereditarily epi--normal if there is a coarser topology such that is hereditarily -normal. A space is hereditarily epi--normal if there is a coarser topology such that is hereditarily -normal.
Note that any Hausdorff hereditarily -normal space is hereditarily epi--normal and any Hausdorff hereditarily -normal space is hereditarily epi--normal. The following is clear.

Corollary 11. A space is hereditarily epi--normal if and only if every subspace of is epi--normal.

Theorem 16. For each topological space , the following cases are equivalent:(1)The space is hereditarily epi--normal.(2)Any subspace in which is an open subspace of a coarser topological space of is epi--normal.(3)For every pair of sets, , of , which are separated in , there exist disjoint open sets , in such that is dense in and is dense in with respect to (we may call this epi--separated).

Proof.   It is clear.  Let , be a pair of sets of which are separated in . Here is an error in set with , . Since and by hypothesis that is -normal, there exist two disjoint open sets , such that But is an open subspace of , and thus and are open in , and since is coarser than , then and are open in as needed.  Let be a subspace of an epi--normal space ; then, there is a coarser topology such that is -normal. It is enough to show that is -normal as a subspace of . Since is , then so is .Let and be a pair of disjoint closed subsets of . Note that and are separated in . By hypothesis, there exist open disjoint subsets , of such that is dense in and is dense in with respect to . Now and are open in , and clearly is dense in and is dense in with respect to the space . That is, is -normal. But is coarser than ; then, is epi--normal as a subspace of . Hence, is hereditarily epi--normal.
It is officious to observe that an analogous of Theorem 17 is not satisfied for epi--normality. Only parts (1) and (2) hold for hereditarily epi--normal spaces. For example with the usual topology is Hausdorff hereditarily -normal space hence is hereditarily epi--normal but does not satisfies condition (3) of Theorem 17 (for details, see [19]).

Theorem 17. For each topological space , the following cases are equivalent:(1)The space is hereditarily epi--normal.(2)Any subspace of in which is an open subspace of a coarser topological space of is epi--normal.

Proof. The implication is clear. For , let be any subspace of an epi--normal space . Then, there is a coarser topology such that is -normal. It is enough to show that the coarser space is -normal as a subspace of . Since is , then so is .
Now let and be any disjoint closed subsets of . Clearly, and . Consider ; then, is an open subspace of with and being disjoint closed subsets of , and it follows that as a subspace of is epi--normal. So, there exist open sets , in such that is dense in and is dense in with respect to the topology , and . Since , we have that and are open disjoint subsets of with being dense in and being dense in with respect to and . That is, is -normal, and hence is epi-hereditarily -normal.

5. Relations with Other Topological Properties

Theorem 18. (1)A space is epinormal if and only if it is epi-mildly normal and the coarser topology of epi-mildly normal is -normal.(2)A space is epinormal if and only if it is epi--normal and the coarser topology of epi--normal is mildly normal.

Proof. We prove (1), and the proof of (2) is similar.
(1) (2) The first direction is clear. Now, we prove the second direction.
Let be epi-mildly normal such that there is a coarser topology in which is mildly normal and -normal; therefore, by Theorem 2.4 in [5], is normal. Hence, is epinormal.
However, epi-mild normality does not imply epi--normality nor epi--normality, and here is an example.

Example 2. Let be the relatively prime integer topology [6]; then, is Hausdorff not completely Hausdorff. Hence, it is not epi--normal space. Also, the space is first countable, so it is not epi--normal. However, is epi-mildly normal space (see [20]).
Recall that a topological space is called partially normal [21] if for any two disjoint subsets and of , where is closed domain and is -closed, there exist two disjoint open subsets and of such that and . The topology on generated by the family of all open domain is denoted by . The space is called the semi-regularization of [8].

Theorem 19. If is a seminormal and partial normal space and is , then is epi--normal.

Proof. It is enough to show that is -normal. Let and be any disjoint closed sets in , then is an open set containing the closed set . By seminormality of , there is an open set such that . Since is regularly open and by using Theorem 2.11 in [20], there exists an open set in such that . Therefore, ; then, is -normal. Hence, is epi--normal.
Note that epi--normality does not imply normality. For example, Either-or Topology [6]. It is normal because the only disjoint closed sets are the ground set and the empty set. But it is not epi--normal because it is not completely Hausdorff. Also, epi--normality does not imply almost normality; a good example for this is right order topology [6], where the topology is generated by basis of the form . This topology is almost normal because there are no nonempty disjoint closed sets. But it is not epi--normal because it is not completely Hausdorff space.
Moreover, epi--normality does not imply partial normality. a basic open neighborhood for the irrational point is isolated, and a basic open neighborhood for a rational point is the same as in the usual topology and consider denote the Michael product, where with the usual topology, see [11]. Since all minimal completely regular spaces are compact, we have the following theorem. Observe that is epi--normal, by taking the coarser topology of . Then, is Hausdorff -normal, since it is metrizable. Since all minimal completely regular spaces are compact [[23], 3.3].

Theorem 20. In the class of minimal Tychonoff spaces, any epi-mildly normal space is epi--normal.
Since any nontrivial Mrówka space is epinormal (see [24]) and it is not -normal (see [5]), we conclude the following.

Theorem 21. Any nontrivial Mrówka space is epi--normal non--normal.
Since any almost normal is almost regular and every almost regular semiregular space is regular, then we have the following theorem.

Theorem 22. In semiregular space, every epi-almost normal space is epi--normal.
Recall that a Hausdorff space is called -closed if is a closed subspace of every Hausdorff containing it (see [[11], 3.12.5]. Since a regular space is -closed if and only if is compact and since every epi--normal is epiregular (see Corollary 1), we will have the following.

Corollary 12. Let be epi--normal compact space; then, the coarser topology of epi--normality is -closed.
Since any regular Lindelöff space is normal, then the following is true.

Corollary 13. Let be epi--normal, and the coarser topology of epi--normality is Lindelöff; then, is epinormal.
Recall that a topological space is called -paracompact [25] if there exist a paracompact space and a bijective function such that the restriction is a homeomorphism for each compact subspace .

Theorem 23. Let be a -paracompact and Fréchet; then, is epi--normal.

Proof. Let be -paracompact and Fréchet; then, is epinormal by Theorem 2.16 in [25], and hence it is epi--normal.

Theorem 24. If is a Hausdorff nearly compact space, then is epi--normal.

Proof. Let be a Hausdorff nearly compact space, and let be the semi-regularization of , then is a Hausdorff nearly compact space. Therefore, is and hence and -normal. Thus, is epi--normal.
Recall that a topological space is called extremally disconnected if the closure of any open subset of is open [22].
Since every -normal, extremely disconnected space is normal [5], then we have the following.

Corollary 14. If is an epi--normal space and the coarser topology of epi--normality is extremely disconnected, then is epinormal.
Epi--normality does not imply extreme disconnectedness. For example, weak parallel topology [6] is epi--normal but not extremely disconnected.

Corollary 15. Any epi--normal zero-dimensional space is semiregular.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest to report regarding the present study.

Acknowledgments

This study was supported by the Taif University Researchers Supporting Project (TURSP-2020/207), Taif University, Taif, Saudi Arabia.