# Epi-α-Normality and Epi-β-Normality

Authors: Nadia Gheith; Samirah ALZahrani
Journal: Journal of Mathematics
 (2021)

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## Abstract

A topological spaceY,τ is called epi-α-normal (epi-β-normal) if there is a coarser topology τ′ on Y such that Y,τ′ is T1α-normal (T1β-normal). We investigate these properties and show some examples to explain the relationships of epi-α-normal (epi-β-normal) with other weaker versions of normality and some topological spaces.


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## Body

## 1. Introduction

In [1], the notions of α-normality and β-normality were investigated in relative sense, and their dependency and independency on other generalizations of relative normality were proved. Some relative generalizations of β-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 β-normal. Also, almost β-normal spaces as a simultaneous generalization of β-normal and almost normal spaces were introduced by Das et al. [2]. Another new generalization of normality, namely, weak β-normality, in terms of θ-closed sets was introduced, which turns out to be a simultaneous generalization of β-normality and θ-normality (see [3]).In this paper, we introduce new topological properties called epi-α-normality and epi-β-normality. We investigate these properties and show some examples to explain the relationships between epi-α-normality (epi-β-normality) with other weaker versions of normality and some topological spaces. We prove that submetrizablity, T4 imply epi-β-normality and T1α-normality implies epi-α-normality, and we prove the converse is not true in general. Throughout this article, a T4 space is a T1-normal space, a completely regular space with T1 is Tychonoff space, and regular space with T1 is T3.


## 2. Epi-α-Normality and Epi-β-Normality

Recall that a topological spaceY,τ is called epinormal [4] if there is a coarser topology τ′ on Y such that Y,τ′ is T4. A topological space Y,τ is called α-normal [5] if for any two disjoint closed subsets F1 and F2 there exist disjoint open subsets H1 and H2 of Y such that F1∩H1 is dense in F1 and F2∩H2 is dense in F2. A topological space Y,τ is called β-normal [5] if for any two disjoint closed subsets F1 and F2 there exist open subsets H1 and H2 of Y such that F1∩H1 is dense in F1, F2∩H2 is dense in F2, and H1¯∩H2¯=∅.Definition 1.
A topological spaceY,τ is called epi-α-normal (epi-β-normal) if there is a coarser topology τ′ on Y such that Y,τ′ is T1α-normal (T1β-normal).
Note, if we assumeY,τ′ to be just α-normal (β-normal) with out assume T1 in the above definition, therefore any space will be epi-α-normal (epi-β-normal) as the indiscret topology.
Obviously, anyT1α-normal space is epi-α-normal, any T1β-normal space is epi-β-normal, and any T4 space is epi-β-normal just by taking τ=τ′, and any epinormal space is epi-β-normal since every normal space is β-normal.
Note that any submetrizable space [4] is epi-β-normal, but the converse is not true in general. For example, ω1+1 is an epi-β-normal space is Hausdorff and compact; therefore, T4. But it is not submetrizable (for details, see [4]).
Epi-α-normality and α-normality do not imply each other. For example, the Niemytzki plane is epi-α-normal since it is submetrizable, but it is not α-normal by the standard proof of nonnormality. On the other hand, an example of an α-normal space which is not epi-α-normal is any indiscrete space which has more than one point.
Observe that if the coarser topologyY,τ′ in Definition 1 is Ti, i∈0,1,2, then so is Y,τ. Since any β-normal (epi-α-normal) is Hausdorff [5], we conclude the following.Theorem 1.
Any epi-β-normal (epi-α-normal) space is Hausdorff.
Note that every topological space which is not Hausdorff can neither be epi-β-normal nor be epi-α-normal. For example, the Double pointed reals topological space defined in [6] is β-normal, and hence α normal but neither epi-β-normal nor epi-α-normal since it is not Hausdorff.Theorem 2.
Every epi-β-normal space is completely Hausdorff.Proof.
LetY,τ be an epi-β-normal space. Let τ′ be a coarser topology on Y; then, Y,τ′ is T1β-normal. We assume that Y has more than one element and take distinct x,y∈Y. Since Y,τ′ is T1β-normal, then it is T2 (see [5]). Choose H1,H2∈τ such that x∈H1, y∈H2, and H1∩H2=∅. Now since Y,τ′ is T1β-normal, then by Proposition 1.1 in [5], Y,τ′ is regular, and therefore choose U1,U2∈τ′ such that x∈U1⊆U1¯τ′⊆H1 and y∈U2⊆U2¯τ′⊆H2. We have U1,U2∈τ and since A¯τ⊆A¯τ′ for any A⊆Y, we get U1¯τ∩U2¯τ=∅.
Note that any space which is not completely Hausdorff cannot be epi-β-normal. For example, Prime Integer topology [6]. In this space, the set Y=ℤ+ of positive integers, where any open neighborhood Uab=b+na∈Y:n∈ℤ, since the closures of any two open neighborhood Uab, Ucd contain in common all multiples of a,c, so it is not completely Hausdorff (see [6]), and by Theorem 2, the space Y cannot be epi-β-normal.
Observe that epi-α-normality may not be necessarily completely Hausdorff. However, we have the following theorem.Theorem 3.
IfY,τ is an epi-α-normal space and the coarser topology of epi-α-normality Y,τ′) is first countable, then Y,τ is completely Hausdorff.Proof.
By the similar argument of Theorem2, since every Hausdorff α-normal first countable is regular, the proof follows (see [7]).
Since everyβ-normal space satisfying T1 axiom is regular (see [5]), we recall that a topological space Y,τ is called epiregular [8] if there is a coarser topology τ′ on Y such that Y,τ′ is T3, so easily we conclude the following.Corollary 1.
Any epi-β-normal space is epiregular.
AnyT2 paracompact space is T4, so we conclude the following.Corollary 2.
IfY,τ is an epi-β-normal space and the coarser topology of epi-β-normality Y,τ′ is paracompact, then Y,τ is T4.
Also, it is well known that anyT2 compact space is T4, so we conclude the following.Corollary 3.
Any epi-β-normal compact space is T4.
Recall that a topological spaceX,τ is called α-regular [9, 10] if for any closed subsets F of Y and y∈Y such that y∉F there exist disjoint open subsets H1 and H2 of Y such that y∈H1 and F∩H2 is dense in F. A topological space X,τ is called epi-α-regular (see [9]) if there is a coarser topology τ′ on Y such that Y,τ′ is T1α-regular. Since every α-normal space satisfying T1 axiom is α-regular (see [9, 10]), we conclude the following.Corollary 4.
Every epi-α-normal space is epi-α-regular.Theorem 4.
IfY is an epi-β-normal pseudocompact space of cardinality less than the continuum and the coarser topology of epi-β-normality is completely regular, then Y is T4.Proof.
LetY be an epi-β-normal pseudocompact space of cardinality less than the continuum and τ′ be a coarser topology on Y where Y,τ′ is T1β-normal. Note that Y,τ′ is pseudocompact. Since any β-normal pseudocompact space is countably compact (see Theorem 2.2 in [5]), Y,τ′ is countably compact Hausdorff. Since it is of cardinality less than the continuum, then Y,τ′ is first countable.
Now, we show thatτ′=τ. Let K be any closed set in Y,τ. Assume that the family ℳ=M=U¯τ:U∈τandK⊆U. Clearly we have K=∩M:M∈ℳ. Since the pseudocompactness is hereditary with respect to closed domain ([11], 3.10. F), M is pseudocompact in Y,τ for every M∈ℳ. Then, M is pseudocompact in Y,τ′ for every M∈ℳ. Since pseudocompact subsets of first countable Tychonoff space are closed [12], M is closed in Y,τ′ for every M∈ℳ. Hence, K is closed in Y,τ′ and then τ′=τ. Thus, Y,τ is T4.Corollary 5.
IfY is an epi-β-normal pseudocompact first countable space and the coarser topology of epi-β-normality is completely regular, then Y is T4.Theorem 5.
IfY is an epi-β-normal space and the coarser topology of epi-β-normality is a dense subspace of a product of metrizable spaces, then Y is epinormal.Proof.
Assume thatY is an epi-β-normal space; then, there is a coarser topology τ′ such that Y,τ′ is T1β-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 Y,τ′ is T1β-normal and mildly normal, then by Theorem 2.4 in [5], Y,τ′ is T1 normal and hence T4. Therefore, Y,τ′ is epinormal.
The following is an example ofα normal Hausdorff non-regular space constructed by Murtinová in [7] where she showed that it is an example of an α normal Hausdorff non-regular space.Example 1 (see [7]).
LetY=ω1+1 and define a topology τ such that ω1 with the ordinal topology is an open subspace and a base in the point ω1 will be the collection:(1)UC=ω1∪α+1:α∈C,where C is a closed unbounded subset of ω1 (Club).
The topologyτ is Hausdorff since it is stronger than the order topology on ω1+1. This space is epi-α-normal since it is α-normal Hausdorff and it is epi-β-normal since it is stronger than the order topology on ω1+1, but it is not β-normal nor regular.
Murtinová in [7] proved that every first countable α-normal Hausdorff space is regular, so we have the following result.Theorem 6.
IfY is epi-α-normal and the coarser topology of epi-α-normality Y,τ′ is first countable, then Y is epiregular.
Recall that a topological spaceY,τ is called epicompletely regular [15] if there is a coarser topology τ′ on Y such that Y,τ′ is Tychonoff.Theorem 7.
If a topological spaceY is epicompletely regular and the coarser topology of epicompletely regularity is second countable (countable), then the space is epi-β-normal.Theorem 8.
If a topological spaceY is epi-β-normal and the coarser topology of epi-β-normal is compact second countable, then the space is submetrizable.
The similar argument is correct for epi-α-normality.


## 3. Dowker’s Characterisation of Epi-α-Normal and Epi-β-Normal

In light of Dowker’s characterisation ofα-normal and β-normal (for details, see [16]), it is natural to ask if this can be satisfied for epi-α-normal and epi-β-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 spaceY is α-normal if and only if for every pair F1 and F2 of disjoint closed subsets of Y there exists an open set U of Y such that U∩F1¯=F1 and U∩F2¯ is nowhere dense in F2.Proof.
LetY,τ be an α-normal space and F1, F2 be disjoint closed subsets of Y. Then, there are two disjoint open subsets U and V such that U∩F1¯=F1 and V∩F2¯=F2. It follows that F2⊆Uc which means that F2∩U=∅. Now let intU¯∩F2¯F2≠∅; then, there is at least an element y∈intU¯∩F2¯F2, so y∈intU¯∩F2¯∩F2, which means that y∈intU¯∩F2 and y∈F2; then, there is an open set W containing y and contained in F2 where W∩U≠∅. This contradicts F2⊆Uc. Therefore, intU¯∩F2¯F2=∅. Hence, U∩F2 is nowhere dense in F2.
Conversely, letF1 and F2 be disjoint closed subsets of Y; there exists an open set U of Y such that U∩F1¯=F1 and U∩F2¯ is nowhere dense in F2. Then, intU¯∩F2¯F2=∅ which implies that intU¯∩F2¯∩F2=∅. Now let y∈F2; then, y∉intU¯∩F2¯. It follows that for any open subset W in F2 containing y, W is not contained in U¯∩F2 and then W∩U¯c∩F2≠∅. Let V=U¯c. Then, V is an open set disjoint from U and y∈V∩F2¯, and so F2=V∩F2¯. Hence, Y,τ is an α-normal space.
The following theorem has the same argument of Theorem 2.3 in [16].Theorem 9.
LetτO be the order topology defined on ω+1. The product Y×ω+1 is epi-α-normal if and only if(1)
Y is epi-α-normal.(2)
LetY,τ′ be the coarser topology of epi-α-normality. If Kn:n∈ω is a family of closed sets in Y,τ′ and K=∩n∈ωclτ′∪i=n∞Ki and C is a closed subset of Y,τ′ disjoint from K, then there is a family Wn:n∈ω of open sets in τ′ such that Wn∩Kn is τ′ dense in Kn and ∩n∈ωclτ′∪i=n∞Wi is τ′ nowhere dense in C.
The case of epi-β-normal spaces is similar to the case of epi-α-normal spaces, but it is more complicated. First, we introduce the following lemma induced from [16].Lemma 2.
LetY be a T1 space. If Y×ω+1 is epi-β-normal, then(1)
Y is β-normal.(2)
LetY,τ′ be the coarser topology of epi-β-normality. If Kn:n∈ω is a family of closed sets in Y,τ′ and K=∩n∈ωclτ′∪i=n∞Ki and C is a closed subset of Y,τ′ disjoint from K, then there is a family Wn:n∈ω of open sets in τ′ such that Wn∩Kn is τ′ dense in Kn and ∩n∈ωclτ′∪i=n∞Wi is disjoint from C.
Recall that a topological space is said to beα-countably paracompact [16] if for every decreasing sequence Fn:n∈ω of closed subsets of Y satisfying ∩n∈ωFn=∅ there exists a sequence Wn:n∈ω of open subsets of Y such that Wn∩Fn is dense in Fn for n∈ω and ∩n∈ωWn¯=∅.
With the definition ofα-countably paracompact from [16] and Lemma 2, it is not hard to prove the following corollary.Corollary 6.
IfY×ω+1 is epi-β-normal, then Y is epi-β-normal and α-countably paracompact.
With Lemma2, we are ready to introduce an important result of epi-β-normal spaces which has the same argument of [16].Theorem 10.
LetτO be the order topology defined on ω+1. The product Y×ω+1 is epi-β-normal if and only if(1)
Y is epi-β-normal.(2)
Condition of Lemma2 is satisfied.(3)
LetY,τ′ be the coarser topology of epi-β-normality. For each decreasing sequence Fn:n∈ω of closed sets in Y,τ′ satisfying ∩n∈ωFn=∅, there is a sequence Vn:n∈ω of open subset of Y,τ′ where Fn⊂Vn for n∈ω and ∩n∈ωclτ′Vn is nowhere dense in the relative topology of F0 with respect to the coarser topology Y,τ′.Corollary 7.
LetY be epi-α-normal and countably paracompact. Then, Y×ω+1 is epi-α-normal.Corollary 8.
LetY be epi-β-normal and countably paracompact. Then, Y×ω+1 is epi-β-normal.


## 4. Properties of Epi-α-Normality and Epi-β-Normality

LetY be any topological space. Let Y′=Y×1. Note that Y∩Y′=∅. Let AY=Y∩Y′. For an element y∈Y, y,1∈AY denoted by y′, and for a subset A⊆Y, we have A′=y′:y∈A=A×1⊆Y′. For each y′∈Y′, let ℬy′=y′. For each y∈Y, let ℬy=U∪U′\y′:Uis open inYandy∈U. We define a basic neighborhood ℬ=ℬy:y∈Y∪ℬy′:y′∈Y′ for a unique topology defined on AY. This generated topological space AY is called Alexandroff duplicate of Y (for revision, see [17]).Theorem 11.
LetY be an epi-α-normal space, and so is its Alexandroff duplicate AY.Proof.
LetY be any epi-α-normal space and let τ be a topology on Y; since Y is epi-α-normal, then there is a coarser topology τ′ on Y such that Y,τ′ is T1α-normal. Since by Theorem 0.1 in [18] α-normality is preserved by the Alexandroff duplicate space and also T1, then AY,S′ is also T1α-normal, and it is coarser than AY,S by the topology of the Alexandroff duplicate. Hence, AY,S is epi-α-normal.
We do not know if the above theorem is valid for epi-β-normal space.Theorem 12.
β-normality and α-normality are additive properties.Proof.
We prove the case forβ-normality, and that for α-normality will be similar.
LetYαα∈Λ be a family of β-normal spaces and A1, A2 be two disjoint closed subsets of the sum ⊕α∈ΛYα. By Proposition 2.2.1 in [11], the intersections A1∩Yα and A2∩Yα are closed in Yα for each α∈Λ. From β-normality of Yα, it follows that there are two open sets Uα and Vα in Yα where(2)A1∩Yα∩Uα¯=A1∩Yα,A2∩Yα∩Vα¯=A2∩Yα,U∩A1¯=∅.
LetU=∪α∈ΛUα and V=∪α∈ΛVα; then, clearly(3)A1∩U¯=∪α∈ΛA1∪Yα¯=∪α∈ΛA1¯=A1¯=A1,A2∩V¯=∪α∈ΛA2∪Yα¯=∪α∈ΛA2¯=A2¯=A2,U¯∩V¯=∪α∈ΛUα¯∩∪α∈ΛVα¯=∪α∈ΛUα¯∩Vα¯=∅.
SinceU and V are open in ⊕α∈ΛYα, the sum ⊕α∈ΛYα is β-normal.Theorem 13.
Epi-β-normality and epi-α-normal space are additive properties.Proof.
LetYα,τα be epi-β-normal spaces for each α∈Λ. Let τα′ be a topology on Yα coarser than τα such that Yα,τα′ is T1β-normal. Since T1and β-normality is also additive by Theorem 12. Then, and the sum ⊕α∈ΛYα,τα′ is T1 and β-normal, and its topology is coarser than the topology on ⊕α∈ΛYα,τα. The case for epi-α-normality is similar.Theorem 14.
LetY be a β-normal (α-normal) space, and h:Y⟶Z is an onto, continuous, open, and closed function. Then, Z is β-normal (α-normal).Proof.
We prove the case forβ-normality, and that for α-normality will be similar. Let A1 and A2 be disjoint closed subsets of Z. Then, by continuity of h, h−1A1 and h−1A2 are disjoint closed subsets of Y. Since Y is a β-normal space [5], there exists an open subset G of Y such that h−1A1∩G¯=h−1A1 and h−1A2∩G¯=∅. Since h−1A2∩G¯=∅, then A2∩hG¯=∅. Observe that hG¯ is a closed set containing the open set hG and hG¯⊆hG¯. Then, hG¯∩A2=∅. Now it is sufficient to show that A1∩hG¯=A1. Let z∈A1 and W be an open set containing z; then, h−1z⊆h−1A1∩h−1W. Since h−1A1∩G¯=h−1A1, h−1A1∩G∩h−1W≠∅. Hence, by surjectivity of h, A1∩hG∩W=hh−1A1∩hG∩hh−1W⊇hh−1A1∩G∩h−1W≠∅ as required.Corollary 9.
LetY,τ be an epi-β-normal (epi-α-normal) space, and g:Y,τ⟶Z,S is an onto, continuous, open, and closed function. Then, Z is epi-β-normal (epi-α-normal).Proof.
We only prove the case for epi-β-normality. Let Y,τ be any epi-β-normal space, and let τ′ be a coarser topology on Y such that Y,τ′ is β-normal, T1. Since g:Y⟶Z is an onto, continuous, open, and closed function, then by Theorem 14Z,S′, where S′=gU:U∈τ′, is β-normal, and it is obviously T1 and S′ is coarser than S. Hence, Z,S is epi-β-normal.Corollary 10.
Epi-β-normality and epi-α-normality are topological properties.Theorem 15.
LetY,τ be an epi-β-normal (epi-α-normal) space, and A,τA′ is a closed subspace of the coarser topology of epi-β-normality (epi-α-normal) Y,τ′; then, the subspace A,τA is epi-β-normal (epi-α-normal).Proof.
This is straightforward since any closed subspace of anβ-normal (α-normal) space is β-normal (α-normal) (see Proposition 1.3 in [5]).Definition 2.
A spaceY,τ is hereditarily epi-α-normal if there is a coarser topology τ′ such that Y,τ′ is T1 hereditarily α-normal. A space Y,τ is hereditarily epi-β-normal if there is a coarser topology τ′ such that Y,τ′ is T1 hereditarily β-normal.
Note that any Hausdorff hereditarilyα-normal space is hereditarily epi-α-normal and any Hausdorff hereditarily β-normal space is hereditarily epi-β-normal. The following is clear.Corollary 11.
A spaceY,τ is hereditarily epi-α-normal if and only if every subspace of Y,τ is epi-α-normal.Theorem 16.
For each topological spaceY,τ, the following cases are equivalent:(1)
The spaceY,τ is hereditarily epi-α-normal.(2)
Any subspaceZ,τZ in which Z,τZ′ is an open subspace of a coarser topological space Y,τ′ of Y,τ is epi-α-normal.(3)
For every pair of sets,A1, A2 of Y,τ, which are separated in Y,τ′, there exist disjoint open sets U1, U2 in Y,τ such that A1∩U1 is dense in A1 and A2∩U2 is dense in A2 with respect to Y,τ′ (we may call this epi-α-separated).Proof.
1⇒2 It is clear.2⇒3 Let A1, A2 be a pair of sets of Y,τ which are separated in Y,τ′. Here is an error in set ZZ=Y,clτ′A1∩clτ′A2 with A1, A2⊆Z. Since clτZ′A1∩clτZ′A2=∅ and by hypothesis that Z,τZ′ is T1α-normal, there exist two disjoint open sets U1, U2⊆Z,τZ′ such that(4)clτZ′U1∩clτZ′A1=clτZ′A1,clτZ′U2∩clτZ′A2=clτZ′A2.ButZ,τZ′ is an open subspace of Y,τ′, and thus U1 and U2 are open in Y,τ′, and since Y,τ′ is coarser than Y,τ, then U1 and U2 are open in Y,τ as needed.3⇒1 Let Z,τZ be a subspace of an epi-α-normal space Y,τ; then, there is a coarser topology τ′ such that Y,τ′ is T1α-normal. It is enough to show that Z,τZ′ is T1α-normal as a subspace of Y,τ′. Since Y,τ′ is T2, then so is Z,τZ′.
LetA1 and A2 be a pair of disjoint closed subsets of Z,τZ′. Note that A1 and A2 are separated in Y,τ′. By hypothesis, there exist open disjoint subsets U1, U2 of Y,τ such that A1∩U1 is dense in A1 and A2∩U2 is dense in A2 with respect to Y,τ′. Now U1∩Z and U2∩Z are open in Z,τZ′, and clearly A∩U1∩Z is dense in A1 and A2∩U2∩Z is dense in A2 with respect to the space Z,τZ′. That is, Z,τZ′ is α-normal. But Z,τZ′ is coarser than Z,τZ; then, Z,τZ is epi-α-normal as a subspace of Y,τ. Hence, Y,τ is hereditarily epi-α-normal.
It is officious to observe that an analogous of Theorem17 is not satisfied for epi-β-normality. Only parts (1) and (2) hold for hereditarily epi-β-normal spaces. For example X=0,1 with the usual topology is Hausdorff hereditarily β-normal space hence is hereditarily epi-β-normal but does not satisfies condition (3) of Theorem 17 (for details, see [19]).Theorem 17.
For each topological spaceY,τ, the following cases are equivalent:(1)
The spaceY,τ is hereditarily epi-β-normal.(2)
Any subspaceA,τA of Y,τ in which A,τA′ is an open subspace of a coarser topological space Y,τ′ of Y,τ is epi-β-normal.Proof.
The implication1⇒2 is clear. For 2⇒1, let A,τA be any subspace of an epi-β-normal space Y,τ. Then, there is a coarser topology τ′ such that Y,τ′ is T1β-normal. It is enough to show that the coarser space A,τA′ is T1β-normal as a subspace of Y,τ′. Since Y,τ′ is T1, then so is A,τA′.
Now letF1 and F2 be any disjoint closed subsets of A,τA′. Clearly, F1=clτA′F1∩A and F2=clτA′F2∩A. Consider Z=Y,clτA′F1∩clτA′F2; then, Z,τZ′ is an open subspace of Y,τ′ with F1 and F2 being disjoint closed subsets of Z,τZ′, and it follows that Z,τZ as a subspace of Y,τ′ is epi-β-normal. So, there exist open sets U1, U2 in Z,τZ such that U1∩F1 is dense in F1 and U2∩F2 is dense in F2 with respect to the topology τZ′, and clτZ′U1∩clτZ′U2=∅. Since A⊆Z, we have that U1∩A and U2∩A are open disjoint subsets of A with F1∩U1∩A being dense in F1 and F2∩U2∩A being dense in F2 with respect to τA′ and clτA′U1∩A∩ClτA′U2∩A=∅. That is, A,τA′ is β-normal, and hence Y is epi-hereditarily β-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 isβ-normal.(2)
A space is epinormal if and only if it is epi-β-normal and the coarser topology of epi-β-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 Y be epi-mildly normal such that there is a coarser topology τ′ in which Y,τ′ is T2 mildly normal and β-normal; therefore, by Theorem 2.4 in [5], Y,τ′ is T2 normal. Hence, Y,τ is epinormal.
However, epi-mild normality does not imply epi-β-normality nor epi-α-normality, and here is an example.Example 2.
Letℕ,ℛPℐ be the relatively prime integer topology ℛPℐ [6]; then, ℕ,ℛPℐ is Hausdorff not completely Hausdorff. Hence, it is not epi-β-normal space. Also, the space ℕ,ℛPℐ is first countable, so it is not epi-α-normal. However, ℕ,ℛPℐ is epi-mildly normal space (see [20]).
Recall that a topological spaceY is called partially normal [21] if for any two disjoint subsets A1 and A2 of Y, where A1 is closed domain and A2 is π-closed, there exist two disjoint open subsets H1 and H2 of Y such that A1⊆H1 and A2⊆H2. The topology on Y generated by the family of all open domain is denoted by τs. The space Y,τs is called the semi-regularization of Y [8].Theorem 19.
IfY,τ is a seminormal and partial normal space and τs is T1, then Y,τ is epi-β-normal.Proof.
It is enough to show thatY,τs is β-normal. Let A1 and A2 be any disjoint closed sets in Y,τs, then Y−A2 is an open set containing the closed set A1. By seminormality of Y,τ, there is an open set U1 such that A1⊆U1⊆intU1¯⊆Y−A2. Since intU1¯ is regularly open and by using Theorem 2.11 in [20], there exists an open set U2 in Y,τs such that A1⊆U2⊆U2¯⊆intU1¯⊆Y−A2. Therefore, A1∩U2¯⊆U2¯⊆intU1¯⊆Y−A2; then, Y,τs is β-normal. Hence, Y,τ is epi-β-normal.
Note that epi-β-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-β-normal because it is not completely Hausdorff. Also, epi-β-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 Pa=y:y>a,a∈ℝ. This topology is almost normal because there are no nonempty disjoint closed sets. But it is not epi-β-normal because it is not completely Hausdorff space.
Moreover, epi-β-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 ℳ×ℙ denote the Michael product, where ℙ with the usual topology, see [11]. Since all minimal completely regular spaces are compact, we have the following theorem. Observe that ℳ×ℙ is epi-β-normal, by taking the coarser topology U×ℙ of ℳ×ℙ. Then, ℳ×ℙ is Hausdorff β-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-β-normal.
Since any nontrivial Mrówka spaceΨA is epinormal (see [24]) and it is not α-normal (see [5]), we conclude the following.Theorem 21.
Any nontrivial Mrówka spaceΨA is epi-α-normal non-α-normal.
Since any almost normalT1 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-β-normal.
Recall that a Hausdorff spaceY is called H-closed if Y is a closed subspace of every Hausdorff containing it (see [[11], 3.12.5]. Since a regular space is H-closed if and only if is compact and since every epi-β-normal is epiregular (see Corollary 1), we will have the following.Corollary 12.
LetY,τ be epi-β-normal compact space; then, the coarser topology of epi-β-normality Y,τ′ is H-closed.
Since any regular Lindelöff space is normal, then the following is true.Corollary 13.
LetY,τ be epi-β-normal, and the coarser topology of epi-β-normality Y,τ′ is Lindelöff; then, Y,τ is epinormal.
Recall that a topological spaceY is called C-paracompact [25] if there exist a paracompact space Z and a bijective function h:Y⟶Z such that the restriction h|K:K⟶hK is a homeomorphism for each compact subspace K⊆Y.Theorem 23.
LetY,τ be a C2-paracompact and Fréchet; then, Y,τ is epi-β-normal.Proof.
LetY,τ be C2-paracompact and Fréchet; then, Y,τ is epinormal by Theorem 2.16 in [25], and hence it is epi-β-normal.Theorem 24.
IfY,τ is a Hausdorff nearly compact space, then Y,τ is epi-β-normal.Proof.
LetY,τ be a Hausdorff nearly compact space, and let τs be the semi-regularization of τ, then τs is a Hausdorff nearly compact space. Therefore, τs is T4 and hence T2 and β-normal. Thus, Y,τ is epi-β-normal.
Recall that a topological spaceY is called extremally disconnected if the closure of any open subset of Y is open [22].
Since everyα-normal, extremely disconnected space is normal [5], then we have the following.Corollary 14.
IfY is an epi-α-normal space and the coarser topology of epi-α-normality is extremely disconnected, then Y is epinormal.
Epi-β-normality does not imply extreme disconnectedness. For example, weak parallel topology [6] is epi-β-normal but not extremely disconnected.Corollary 15.
Any epi-β-normal zero-dimensional space is semiregular.



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*Source: 9311004-2021-07-20.xml*