Some topological properties on C-



α




-Normality and C-



β




-Normality

Samirah Alzahrani

doi:10.1016/j.jksus.2022.102449

Some topological properties on C- $α$ -Normality and C- $β$ -Normality

Samirah Alzahrani^⁎

Department of Mathematics, College of Science, Taif University, Saudi Arabia

⁎Address: Department of Mathematics, College of science, Taif University, P.O.Box 11099, Taif 21944, Saudi Arabia. mam_1420@hotmail.com (Samirah Alzahrani) Samar.alz@tu.edu.sa (Samirah Alzahrani)

Received: 2022-9-23, Accepted: 2022-11-12,

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Peer review under responsibility of King Saud University.

Abstract

A topological space $(Y, τ)$ is called C- $α$ -normal (C- $β$ -normal) if there exist a bijective function $g$ from $Y$ onto $α$ -normal ( $β$ -normal) space $Z$ such that the restriction map $g_{|_{B}}$ from $B$ onto $g (B)$ is a homeomorphism for any compact subspace $B$ of $Y$ . We discuss some relationships between C- $α$ -normal (C- $β$ -normal) and other properties.

Keywords

Normal

α-normal

β-normal

C-normal

Epinormal

Mildly normal

54D15

54B10

1 Introduction

In 2017 we discuss the topological property “ $C$ -normal” (AlZahrani and Kalantan, 2017). In this paper we introduce a new property called C- $α$ -Normality and C- $β$ -Normality. We show any $α$ -normal ( $β$ -normal) space is C- $α$ -normal (C- $β$ -normal), but the converse is not true in general. And we show that any C-normal, lower compact, epinormal, epi- $α$ -normal and epi- $β$ -normal spaces is C- $α$ -normal (C- $β$ -normal). We prove any locally compact is C- $α$ -normal (C- $β$ -normal) but the converse is not true in general. Also observe that a witness function of C- $α$ -normal (C- $β$ -normal) not necessarily to be continuous in general, but it will be continuous under some conditions.

2 C– $α$ -Normality and C- $β$ -Normality

Recall that a topological space $(Y, τ)$ is called an $α$ -normal space (Arhangel’skii and Ludwig, 2001) if for every-two disjoint closed subsets $F$ and $E$ of $Y$ there are two open subsets $G$ and $W$ of $Y$ such that $F \cap G$ is dense in $F$ ; $E \cap W$ is dense in $E$ , and $G \cap W = \emptyset$ , and a topological space $(Y, τ)$ is called a $β$ -normal space (Arhangel’skii and Ludwig, 2001) if for every-two disjoint closed subsets $F$ and $E$ of $Y$ there are two open subsets $G$ and $W$ of $Y$ such that $F \cap G$ is dense in $F$ ; $E \cap W$ is dense in $E$ , and $\bar{G} \cap \bar{W} = \emptyset$ . A topological space $(Y, τ)$ is called C-normal (AlZahrani and Kalantan, 2017) if there exist a bijective function $g$ from $Y$ onto a normal space $Z$ such that the restriction map $g_{|_{B}}$ from $B$ onto $g (B)$ is a homeomorphism for any compact subspace $B$ of $Y$ .

Definition 1.1

A topological space $(Y, τ)$ is called C- $α$ -normal (C- $β$ -normal) if there exist a bijective function $g$ from $Y$ onto $α$ -normal ( $β$ -normal) space $Z$ such that the restriction map $g_{|_{B}}$ from $B$ onto $g (B)$ is a homeomorphism for any compact subspace $B$ of $Y$ .

In these definition, we call the space $Z$ a witness of C- $α$ -normal (C- $β$ -normal) and the function $g$ is called a witness function.

A topological space $(Y, τ)$ is called $α$ -regular (Alzahrani, 2022) if for any $x \in Y$ and a closed subset $A \subset Y$ such that $x \notin A$ there are two disjoint open sets $G$ ; $H \subset Y$ such that $x \in G$ and $\bar{A \cap H} = A$ . And topological space $(Y, τ)$ is called almost $α$ -regular (Alzahrani, 2022) if for any $x \in Y$ and a regular closed subset $A \subset Y$ such that $x \notin A$ there are two disjoint open sets $G$ ; $H \subset Y$ such that $x \in G$ and $\bar{A \cap H} = A$ .

Lemma 1.2

Any regular space is $α$ -regular.

Proof

Let $(Y, τ)$ be a regular space. Pick $y \in Y$ and $F \subseteq Y$ be a closed set such that $y \notin F$ , then there exist two disjoint open sets $W_{1}$ and $W_{2}$ subsets of $Y$ where $y \in W_{1}$ and $F \subseteq W_{2}$ , hence $\bar{F \cap W_{2}} = F$ (note that $\bar{F} = F$ since $F$ is closed), and $W_{1} \cap W_{2} = \emptyset$ , therefore $(Y, τ)$ is $α$ -regular space.

Lemma 1.3

(Alzahrani, 2022) Any $α$ -regular space is almost $α$ -regular.

From Lemma 1.2. and Lemma 1.3. we conclude the following corollary.

Corollary 1.4

Any regular space is almost $α$ -regular.

Lemma 1.5

Any normal space is $α$ -normal.

Lemma 1.6

Lemma 1.6 (Arhangel’skii and Ludwig, 2001)

Any normal space is $β$ -normal.

Proof

Let $Y$ be a normal space. Pick two disjoint closed sets $F_{1}$ and $F_{2}$ subsets of $Y$ . Since $Y$ is normal, then there exist two disjoint open sets $W_{1}$ and $W_{2}$ subsets of $Y$ where $F_{1} \subseteq W_{1}$ , $F_{2} \subseteq W_{2}$ and $W_{1} \cap W_{2} = \emptyset$ . Hence $\bar{F_{1} \cap W_{1}} = F_{1}$ and $\bar{F_{2} \cap W_{2}} = F_{2}$ . It remains to prove $\bar{W_{1}} \cap \bar{W_{2}} = \emptyset$ . For a normal space Y, if F is a closed set, U is an open set and F ⊆ U, then there exist an open set V such that F ⊆ V ⊆ $\bar{V}$ ⊆ U. Now apply this to $F_{1}$ and set $W_{1}$ = V and $W_{2}$ = Y\ $\bar{V}$ .

So we have the following theorem.

Theorem 1.7

Any C-normal space is C- $α$ -normal (C- $β$ -normal).

The converse is true under some conditions, first we mention some definition.

A Hausdorff space $Y$ is extremally disconnected (Engelking, 1977) if the closure of any open set in $Y$ is open. A topological space is called mildly normal (Shchepin, 1972) if any two disjoint regular closed subsets can be separated.

Theorem 1.8

Theorem 1.8 (Arhangel’skii and Ludwig, 2001)

Any $α$ -normal extremally disconnected space is normal.

Proof

Let $Y$ be a $α$ -normal extremally disconnected space. Pick two disjoint closed sets $F_{1}$ and $F_{2}$ subsets of $Y$ . Since $Y$ is $α$ -normal, then there exist two disjoint open sets $W_{1}$ and $W_{2}$ subsets of $Y$ where $\bar{F_{1} \cap W_{1}} = F_{1}$ and $\bar{F_{2} \cap W_{2}} = F_{2}$ . Hence $F_{1} \subseteq \bar{W_{1}}$ and $F_{2} \subseteq \bar{W_{2}}$ . However, $\bar{W_{1}}$ ∩ $W_{2}$ = ∅ since $W_{2}$ is open and $W_{1}$ ∩ $W_{2}$ = ∅. Thus, $\bar{W_{1}}$ ∩ $\bar{W_{2}}$ = ∅ as well since $\bar{W_{1}}$ is open (by extremally disconnectedness) and $\bar{W_{1}}$ ∩ $W_{2}$ = ∅.

Therefore $Y$ is normal space.

From Theorem 1.8, we have the following.

Theorem 1.9

If $Y$ is $C$ - $α$ -normal ( $C$ - $β$ -normal) such that the witness of $C$ - $α$ -normal ( $C$ - $β$ -normal) is extremally disconnected, then $Y$ is $C$ -normal.

Theorem 1.10

If $Y$ is $C$ - $β$ -normal such that the witness of $C$ - $β$ -normal is mildly normal, then $Y$ is $C$ -normal.

Proof

Let $Y$ be C- $β$ -normal. Then the codomain $Z$ witness of C- $β$ -normal is $β$ -normal. Let $F_{1}$ and $F_{2}$ be any disjoint closed subsets of $Z$ . Since $Z$ is $β$ -normal, there exist open subsets $W_{1}$ and $W_{2}$ of $Z$ where $\bar{W_{1}} \cap \bar{W_{2}} = \emptyset$ , $\bar{F_{1} \cap W_{1}} = F_{1}$ and $\bar{F_{2} \cap W_{2}} = F_{2}$ . So $\bar{W_{1}}$ , $\bar{W_{2}}$ are disjoint regular closed subsets containing $F_{1}$ and $F_{2}$ respectively. Since $Z$ is mildly normal, there exist disjoint open subsets $U_{1}$ and $U_{2}$ of $Z$ where $F_{1} \subseteq \bar{W_{1}} \subseteq U_{1}$ and $F_{2} \subseteq \bar{W_{2}} \subseteq U_{2}$ . Hence $Z$ is normal.

Lemma 1.11

Any $α$ -normal space satisfying $T_{1}$ axiom is Hausdorff.

Proof

Let $Y$ be any $α$ -normal $T_{1}$ -space. Let $y, z$ be any two distinct elements in $Y$ . Hence ${y}$ and ${z}$ are disjoint closed subsets of $Y$ , by $α$ -normality, there exist two disjoint open subsets $G_{1}$ and $G_{2}$ of $Y$ where $\bar{{y} \cap G_{1}} = {y}$ and $\bar{{z} \cap G_{2}} = {z}$ which implies $y \in G_{1}$ and $z \in G_{2}$ . Therefore $Y$ is Hausdorff.

Lemma 1.12

Lemma 1.12 (Arhangel’skii and Ludwig, 2001)

Any $β$ -normal space satisfying $T_{1}$ axiom is regular (hence Hausdorff).

By Corollary 1.4. we have the following result.

Corollary 1.13

Any $β$ -normal space satisfying $T_{1}$ axiom is almost $α$ -regular.

Also by Lemma 1.12 and Lemma 1.2 we have the following result.

Lemma 1.14

Any $β$ -normal space satisfying $T_{1}$ axiom is $α$ -regular.

Corollary 1.15

Any $α$ -normal space satisfying $T_{1}$ axiom is $α$ -regular.

By Lemma 1.3. we conclude the following corollary.

Corollary 1.16

Any $α$ -normal space satisfying $T_{1}$ axiom is almost $α$ -regular.

Proposition 1.17

(Murtinov́a, 2002) Every first countable $α$ -normal Hausdorff space is regular.

Recall that a topological space (Y, τ) is called submetrizable (AlZahrani and Kalantan, 2017) if there exists a metric d on Y such that the topology τ d on Y generated by d is coarser than τ.

Theorem 1.18

Every submetrizable space is C- $α$ -normal (C- $β$ -normal).

Proof

Let $(Y, τ)$ be a submetrizable space, the there exists a metrizable $τ^{'}$ such that $τ^{'} \subseteq τ$ . Hence $(Y, τ^{'})$ is $α$ -normal since it is normal, and the identity function $i d_{Y}$ from $(Y, τ)$ onto $(Y, τ^{'})$ is a one-to-one and continuous function. If we take $B$ any compact subspace of $(Y, τ)$ , then $i d_{Y} (B)$ is hausdorff, since it is subspace of $(Y, τ^{'})$ , and by (Engelking, 1977);3.1.13]; $i d_{Y_{|_{B}}}$ is a homeomorphism.

Example 1.19

The Rational Sequence Topology $(R, R S)$ (Steen and Seebach, 1995) is submetrizable being finer than the usual topology $(R, U)$ , so $(R, R S)$ is C- $α$ -normal (C- $β$ -normal).

The converse of Theorem 1.18. is not true in general, for example $ω_{1} + 1$ is C- $α$ -normal (C- $β$ -normal) which is not submetrizable.

Apparently, any $α$ -normal ( $β$ -normal) space is C- $α$ -normal (C- $β$ -normal), to prove this, just by considering $Z = Y$ and $g$ is the identity function.

While in general the converse is not true. We provide some examples below.

Example 1.20

The Half-Disc topological space (Steen and Seebach, 1995) is C- $α$ -normal (C- $β$ -normal) because it is submetrizable by Theorem 1.18. but it is not $α$ -normal nor $β$ -normal because it is first countable and Hausdorff but not regular, so by Proposition 1.17. the Half-Disc topological space is not $α$ -normal space, hence not $β$ -normal. In general C- $α$ -normality (C- $β$ -normality) do not imply $α$ -normality ( $β$ -normality) even with Hausdorff or first countable properties.
The Deleted Tychonoff Plank (Steen and Seebach, 1995), it is C- $α$ -normal (C- $β$ -normal) since it is locally compact by Theorem 2.7. but it is not $α$ -normal nor $β$ -normal see (Arhangel’skii and Ludwig, 2001).
The Dieudonn $\overset{Â ́}{e}$ Plank (AlZahrani and Kalantan, 2017), in example 1.10 we proved that it is C-normal, hence it is C- $α$ -normal (C- $β$ -normal) by Theorem 1.9. but it is not $α$ -normal nor $β$ -normal see (Arhangel’skii and Ludwig, 2001), also not locally compact, hence this example also shows that the converse of Theorem 2.7. is not true.
The Sorgenfrey line square $S \times S$ see (Steen and Seebach, 1995) is not normal, but it is submetrizable space being it is finer than the usual topology on $R \times R$ , so by Theorem 1.18. it is C- $α$ -normal (C- $β$ -normal).

Theorem 1.21

If $Y$ is a compact non– $α$ -normal (non– $β$ -normal) space, then $Y$ can not be C- $α$ -normal (C- $β$ -normal).

Proof

Assume $Y$ is a compact non– $α$ -normal (non– $β$ -normal) space. Suppose $Y$ is C- $α$ -normal (C- $β$ -normal), then there exists $α$ -normal ( $β$ -normal) space $Z$ and a bijective function $g : Y \to Z$ where the restriction map $g_{|_{B}}$ from $B$ onto $g (B)$ is a homeomorphism for any compact subspace $K$ of $Y$ . As $Y$ is compact, then $Y ≅ Z$ , and we have a contradiction as $Z$ is $α$ -normal ( $β$ -normal) while $Y$ is not. Hence $Y$ can not be C- $α$ -normal (C- $β$ -normal).

Observe that a function $g : Y \to Z$ witnessing of C- $α$ -normal ( $C$ - $β$ -normal) of $Y$ not necessarily to be continuous in general, and here is an example.

Example 1.22

Let $R$ with the countable complement topology $C C$ (Steen and Seebach, 1995). We know $(R, C C)$ is $T_{1}$ and the only compact sets are finite, hence the compact subspaces are discrete. If we let $D$ be the discrete topology on $R$ , then obviously the identity function from $(R, C C)$ onto $(R, D)$ is a witnessing of the C- $α$ -normality (C- $β$ -normality) which is not continuous.

But it will be continuous under some conditions as the following theorems.

Theorem 1.23

If $(Y, τ)$ is a C- $α$ -normal (C- $β$ -normal) and Fréchet space, then any function witnessing of C- $α$ -normality (C- $β$ -normality) is continuous.

Proof

Let $Y$ be a Fréchet C- $α$ -normal (C- $β$ -normal) space and $g : Y \to Z$ be a witness of the C- $α$ -normality (C- $β$ -normality) of $Y$ . Let $A \subseteq Y$ and pick $z \in g (\bar{A})$ . There is a unique $y \in Y$ where $g (y) = z,$ thus $y \in \bar{A}$ . since $Y$ is Fréchet, then there exists a sequence $(a_{n}) \subseteq A$ where $a_{n} \to y$ . As the subspace $K = {y} \cup \{a_{n} : n \in N\}$ of $Y$ is compact, the induced map $g_{|_{K}} : K \to g (K)$ is a homeomorphism. Let $U \subseteq Z$ be any open neighborhood of $z$ . Then $U \cap g (K)$ is an open neighborhood of $z$ in the subspace $g (K)$ . Since $g_{|_{K}}$ is a homeomorphism, then $g^{- 1} (U \cap g (K)) = g^{- 1} (U) \cap K$ is an open neighborhood of y in K, then there exists $m \in N$ where $a_{n} \in g^{- 1} (U \cap g (K))$ $\forall n \geq m,$ hence $g (a_{n}) \in (U \cap g (K)) \forall n \geq m,$ then $U \cap g (A) \neq \emptyset .$ Hence $z \in \bar{g (A)}$ and $g (\bar{A}) \subseteq \bar{g (A)} .$ Thus $g$ is continuous.

Since any first countable space is Fréchet, we conclude that, In C- $α$ -normality (C- $β$ -normality) first countable space a function $g : Y \to Z$ is a witness of the C- $α$ -normality (C- $β$ -normality) of $Y$ is continuous. Also, by theorem (Engelking, 1977),3.3.21], we conclude the following.

Corollary 1.24

If $Y$ is a C- $α$ -normal (C- $β$ -normal) $k$ -space and $g$ is a witness function of the C- $α$ -normality (C- $β$ -normality), then $g$ is continuous.

For simplicity, let us call a $T_{1}$ space which satisfies that the only compact subspaces are the finite subsets F-compact. Clearly F-compactness is a topological property.

Theorem 1.25

If $Y$ is F-compact, then $Y$ is $C$ - $α$ -normal ( $C$ - $β$ -normal).

Proof

Let $Y$ be a F-compact. Let $Z = Y$ and let $Z$ with the discrete topology. Hence the identity function from $Y$ onto $Z$ does the job.

Example 1.26

Consider $(R, C C)$ , where $C C$ is the countable complement topology (Steen and Seebach, 1995). We know $(R, C C)$ is $T_{1}$ and the only compact sets are finite, therefore, by Theorem 1.25. $(R, C C)$ is $C$ - $α$ -normal ( $C$ - $β$ -normal). This a fourth example of $C$ - $α$ -normal ( $C$ - $β$ -normal) but not $α$ -normal (nor $β$ -normal).

Notice that any topology finer than a $T_{1}$ topological space is $T_{1}$ . Also any compact sub set of a topological space $(Y, τ)$ is compact in any topology coarser than $τ$ on $Y$ .

Hence any topology finer than F-compact topological space is also F-compact. As an example, $(R, τ)$ denotes the Fortissimo topology on $R$ , see [14, Example 25]. We know that $(R, τ)$ is finer than $(R, C C)$ which is F-compact, hence $(R, τ)$ F-compact too. Thus, $(R, τ)$ is $C$ - $α$ -normal ( $C$ - $β$ -normal).

Theorem 1.27

C- $α$ -normality (C- $β$ -normality) is a topological property.

Proof

Let $Y$ be a C- $α$ -normal (C- $β$ -normal) space and let $Y ≅ W$ . Let $Z$ be a $α$ -normal ( $β$ -normal) space and let $g : Y \to Z$ be a bijective function where the restriction map $g_{|_{B}}$ from $B$ onto $g (B)$ is a homeomorphism for any compact subspace $B \subseteq Y$ . Let $k : W \to Y$ be a homeomorphism. Hence $Z$ and $g ° k : W \to Z$ satisfy the requirements.

3 C– $α$ -Normality (C- $β$ -Normality) and some other properties

Definition 2.1

A topological space $(Y, τ)$ is called C- $α$ -regular if there exists a bijective function $g$ from $Y$ onto $α$ -regular space $Z$ such that the restriction map $g_{|_{B}}$ from $B$ onto $g (B)$ is a homeomorphism for any compact subspace $B$ of $Y$ .

This definition is new and we will study some of its properties later.

Corollary 2.2

If $Y$ is C- $α$ -normal (C- $β$ -normal) space and the witness of the C- $α$ -normality (C- $β$ -normality) of $Y$ is $T_{1}$ , then $Y$ is C- $α$ -regular.

We prove this corollary by Lemma 1.11, Lemma 1.12. We defined C-regular in (AlZahrani, 2018).

Corollary 2.3

If $Y$ is C- $β$ -normal space and the codomain witness of the C- $β$ -normality of $Y$ is $T_{1}$ , then $Y$ is C-regular.

We prove this corollary by Lemma 1.12.

Corollary 2.4

If $Y$ is a C- $α$ -normal (C- $β$ -normal) Fréchet space and the witness of the C- $α$ -normality (C- $β$ -normality) is $T_{1}$ , then $Y$ is $T_{2}$ .

Proof

Let $Y$ is a C- $α$ -normal (C- $β$ -normal) Fréchet space, then there exist $α$ -normal ( $β$ -normal) space $Z$ (witness of the C- $α$ -normality (C- $β$ -normality)) and a bijective function $g : Y \to Z$ such that the restriction map $g_{|_{B}}$ from $B$ onto $g (B)$ is a homeomorphism for any compact subspace $B$ of $Y$ , then by Theorem 1.23. $g$ is continuous. Let any $a, b \in Y$ be such that $a \neq b$ , then $g (a) \neq g (b)$ , $g (a), g (b) \in Z$ . Since $Z$ is $α$ -normal ( $β$ -normal) and $T_{1}$ , then by Lemma 1.11 (Lemma 1.12) the space $Z$ is $T_{2}$ , then there exist $W_{1}$ and $W_{2}$ are open sets in $Z$ where $g (a) \in W_{1}, g (b) \in W_{2}$ and $W_{1} \cap W_{2} = \emptyset$ . Since $W_{1}, W_{2}$ are open sets in $Z$ and $g$ is continuous, then $g^{- 1} (W_{1})$ and $g^{- 1} (W_{2})$ are open sets in $Y$ , $a \in {g^{- 1} (W}_{1}), b \in {g^{- 1} (W}_{2})$ and ${g^{- 1} (W}_{1}) \cap {g^{- 1} (W}_{2}) = {g^{- 1} (W}_{1} \cap W_{2}) = \emptyset$ . Hence $Y$ is $T_{2}$ .

Theorem 2.5

Any C-regular Fréchet Lindelof space is C- $α$ -normal (C- $β$ -normal).

Proof

Let $Y$ be any C-regular Fréchet Lindelof space. Let $Z$ be a regular space and $g : Y \to Z$ be a continuous bijective function see Theorem 1.23. By (Engelking, 1977); 3.8.7] $Z$ is Lindelof. Since any regular Lindelof space is normal (Engelking, 1977), 3.8.2]. Hence $Y$ is C- $α$ -normal (C- $β$ -normal).

C- $α$ -normality (C- $β$ -normality) does not imply C- $α$ -regularity nor C-regular, for example.

Example 2.6

Consider the real numbers set $R$ with its right ray topology $R$ , where $R = {\emptyset, R} \cup {(b, \infty) : b \in R}$ . As any two non-empty closed sets must be intersect in $(R, R)$ , then it is normal, and by Lemma in above, it is $α$ -normal ( $β$ -normal), hence C- $α$ -normal (C- $β$ -normal). Now, suppose that $(R, R)$ is C- $α$ -regular. Take $α$ -regular space $Z$ and a bijective function $g$ from $R$ onto $Z$ where the restriction map $g_{|_{B}}$ from $B$ onto $g (B)$ is a homeomorphism for any compact subspace $B$ of $R$ . We know that a subspace $B$ of $(R, R)$ is compact if and only if $B$ has a minimal element. Hence $[1, \infty)$ is compact, then $g_{|_{[1, \infty)}} : [1, \infty) \to g ([1, \infty)) \subset Z$ is a homeomorphism, it means $[1, \infty)$ as a subspace of $(R, R)$ is $α$ -regular which is a contradiction, since $[1, 4]$ is closed in subspace $[1, \infty)$ and $4.5 \notin [1, 4]$ , but any non-empty open sets on $[1, \infty)$ must intersect. Then $(R, R)$ cannot be C- $α$ -regular (C-regular).

Recall that a topological space $(Y, τ)$ is called Locally Compact (AlZahrani and Kalantan, 2017) if $(Y, τ)$ is Hausdorff and for every $y \in Y$ and every open neighborhood $V$ of $y$ there exists an open neighborhood $U$ of $y$ such that $y \in U \subseteq \bar{U} \subseteq V$ and $\bar{U}$ is compact.

Theorem 2.7

Every locally compact space is C- $α$ -normal (C- $β$ -normal).

Proof

Let $Y$ be locally compact space. By (Engelking, 1977), 3.3.D], there exists $T_{2}$ compact space $Z$ and hence $α$ -normal ( $β$ -normal), and a continuous bijective function $g : Y \to Z$ . We have $g_{|_{K}}$ from $K$ onto $g (K)$ is a homeomorphism for any compact subspace $K$ of $Y$ , because continuity ,1–1 and onto are inherited by g, also $g_{|_{K}}$ is closed since $K$ is compact and g(K) is $T_{2}$ .

Example 2.8

Consider $ω_{1}$ , the first uncountable ordinal, we consider $ω_{1}$ as an open subspace of its successor $(ω_{1} + 1)$ , which is compact and hence is locally compact [14, Example 43]. Thus, $ω_{1}$ is locally compact as an open subspace of a locally compact space, see (Engelking, 1977),3.3.8]. Then by Theorem 2.7. $ω_{1}$ is C- $α$ -normal (C- $β$ -normal).

The converse of Theorem 2.7. is not true in general. We introduce the following example of C- $α$ -normal (C- $β$ -normal) which is not locally compact.

Example 2.9

Consider the quotient space $R / N$ . Let $Z = (R \ N) \cup {i}$ , where $i = \sqrt{- 1}$ . Define $g : R \to Z$ as follows:

g (a) = \{\begin{matrix} \begin{matrix} a & for & a \in R \ N \end{matrix} \\ i \begin{matrix} for & a \in N \end{matrix} \end{matrix}

Now consider $R$ with the usual topology $U$ . Define the topology $τ = {V \subseteq Z : g^{- 1} (V) \in U}$ on $Z$ . Then $g : (R, U) \to (Z, τ)$ is a closed quotient mapping. We explain the open neighborhoods of any element in $Z$ as follows: The open neighborhoods of each $a \in R \ N$ are $(a - ε, a + ε) \ N$ where $ε$ is a natural number. The open neighborhoods of $i \in Z$ are $(G \ N) \cup {i}$ , where $G$ is an open set in $(R, U)$ such that $N \subseteq G$ . It is clear that $(Z, τ)$ is $T_{3}$ , but it is not locally compact . $(Z, τ)$ is a continuous image of $R$ with its usual topology, so it is Lindelof and $T_{3}$ , then $(Z, τ)$ is $T_{4}$ . Hence it is C- $α$ -normal (C- $β$ -normal).

A topological space $(Y, τ)$ is called Epi- $α$ -normal (Gheith and AlZahrani, 2021) if there is a coarse topology $τ^{'}$ on $Y$ such that $(Y, τ^{'})$ is $α$ -normal and $T_{1}$ . A topological space $(Y, τ)$ is called Epi- $β$ -normal (Gheith and AlZahrani, 2021) if there is a coarse topology $τ^{'}$ on $Y$ such that $(Y, τ^{'})$ is $β$ -normal and $T_{1}$ . We defined Epinormal in (AlZahrani and Kalantan, 2016). By the same argument of Theorem 1.18. we can prove the following corollary.

Corollary 2.10

Every epinormal space is C- $α$ -normal (C- $β$ -normal).

Corollary 2.11

Every epi- $α$ -normal (epi- $β$ -normal)space is C- $α$ -normal(C- $β$ -normal).

Any indiscrete space which has more than one point is an example of a C- $α$ -normal (C- $β$ -normal) space which is not epi- $α$ -normal (epi- $β$ -normal).

The converse of Corollary 2.9 is true with Fréchet property.

Theorem 2.12

Any C- $α$ -normal (C- $β$ -normal) Fréchet space is epi- $α$ -normal (epi- $β$ -normal).

Proof

Let $(Y, τ)$ be any C- $α$ -normal (C- $β$ -normal) Fréchet space. Let $(Z, τ')$ be $α$ -normal ( $β$ -normal) and $g : (Y, τ) \to (Z, τ')$ be a bijective function. Since $Y$ is Fréchet, $g$ is continuous (see Theorem 1.23). Define $τ^{*} = {g^{- 1} (V) : V \in τ'}$ . Obviously, $τ^{*}$ is a topology on $Y$ coarser than $τ$ such that $g : (Y, τ^{*}) \to (Z, τ')$ is continuous. Also $g$ is open, since if we take $U \in τ^{*}$ , then $U = g^{- 1} (V)$ where $V \in τ'$ . Thus $g (U) = g (g^{- 1} (V)) = V$ which gives that $g$ is open. Therefore $g$ is a homeomorphism. Thus $(Y, τ^{*})$ is $α$ -normal ( $β$ -normal). Hence $(Y, τ)$ is epi- $α$ -normal (epi- $β$ -normal).

A topological space $(Y, τ)$ is called lower compact (Kalantan et al., 2019) if there exists a coarser topology $τ^{'}$ on $Y$ such that $(Y, τ^{'})$ is $T_{2}$ -compact.

Theorem 2.13

Any lower compact space is C- $α$ -normal (C- $β$ -normal).

Proof

Let $(Y, τ)$ is lower compact, then $(Y, τ^{'})$ is $T_{2}$ -compact, hence normal and the identity function $i d_{Y} : (Y, τ) \to (Y, τ^{'})$ is a continuous and bijective. If we take $B$ any compact subspace of $(Y, τ)$ , then ${i d_{Y}}_{|_{B}}$ is a homeomorphism by (Engelking, 1977);3.1.13].

In general, the converse of Theorem 2.13. is not true, for example consider a countable complement topology on an uncountable set, it is C- $α$ -normal (C- $β$ -normal) since it is F-compact, but it is not lower compact because it is not $T_{2}$ .

Theorem 2.14

If $(Y, τ)$ is C- $α$ -normal compact Fréchet space and the witness of the C- $α$ -normality is $T_{1}$ , then $(Y, τ)$ is lower compact.

Proof

Pick $α$ -normal space $(Z, τ^{*})$ and a bijective function $g : (Y, τ) \to (Z, τ^{*})$ such that $g_{|_{B}} : B \to g (B)$ is a homeomorphism for any compact subspace $B \subseteq Y$ . Since $Y$ is Fréchet, then $g$ is continuous. Hence $(Z, τ^{*})$ is compact. Since $(Z, τ^{*})$ is $T_{1}$ $α$ -normal space, then by Lemma 1.11. it is Hausdorff. Hence $(Z, τ^{*})$ is $T_{2}$ compact. Define a topology $τ^{'}$ on $Y$ as follows $τ^{'} = \{g^{- 1} (V) : V \in τ^{*}\} .$ Then $τ^{'}$ is coarser than $τ$ and $g : (Y, τ^{'}) \to (Z, τ^{*})$ is a bijection continuous function. Let any $U \in τ^{'}$ , then $U$ is of the form $g^{- 1} (V)$ for some $V \in τ^{*}$ . Hence $g (U) = g (g^{- 1} (V)) = V$ . Thus $g$ is open. Hence $g$ is a homeomorphism. So $(Y, τ^{'})$ is $T_{2}$ compact. Therefore $(Y, τ)$ is lower compact.

Theorem 2.15

If $(Y, τ)$ is C- $β$ -normal compact Fréchet space and the witness of the C- $β$ -normality is $T_{1}$ , then $(Y, τ)$ is lower compact.

4 Conclusion

The aim of this paper is to introduce a new weaker version of normality called C- $α$ -normal and C- $β$ -normal. We show that some relationships between this a new topological property and some other topological properties, and there are still many topological properties that the researcher can study in this topic.

Acknowledgments

This research received funding from Taif University Researchers Supporting Project number (TURSP-2020/207), Taif University, Taif, Saudi Arabia.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References

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Appendix A

Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.jksus.2022.102449.

Appendix A

Supplementary data

The following are the Supplementary data to this article:

Supplementary data 1

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Introduction

C– α -Normality and C- β -Normality

C– α -Normality (C- β -Normality) and some other properties

Conclusion

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[1] AlZahrani S., . C-Regular Topological Spaces. J. Mathemat. Ana.. 2018;9(3):141-149.
[Google Scholar]

[2] Alzahrani S., . Almost α-regular spaces. Journal of King Saud University-Science.. 2022;34(1)
[Google Scholar]

[3] AlZahrani S., Kalantan L., . Epinormality. J. Nonlinear Sci. Appl.. 2016;9:5398-5402.
[Google Scholar]

[4] AlZahrani S., Kalantan L., . C-Normal Topological Property. Filomat.. 2017;31(2):407-411.
[Google Scholar]

[5] Arhangel’skii, A., Ludwig L. D., 2001. On α-Normal and β-Normal Spaces, Comment. Math. Univ. Carolinae. 42(3),507-519.

[6] Engelking R., . General Topology. Warszawa: PWN; 1977.

[7] Gheith N., AlZahrani S., . Epi-α-Normality and Epi-β-Normality. J. Math. (Wuhan). 2021;3:1-7.
[Google Scholar]

[8] Kalantan L., Saeed M.M., Alzumi H., . C-Paracompactness and C₂-Paracompactness. Turk. J. Math.. 2019;43(1):9-20.
[Google Scholar]

[9] Murtinov́a E., . A β-Normal Tychonoff Space Which is Not Normal. Comment. Math. Univ. Carolinae.. 2002;43(1):159-164.
[Google Scholar]

[10] Shchepin E.V., . Real functions and spaces that are nearly normal, siberian Math. J.. 1972;13:820-829.
[Google Scholar]

[11] Steen L., Seebach J.A., . Countrexample in Topology. INC, New York: Dover Publications; 1995.

Some topological properties on C- α -Normality and C- β -Normality

Abstract

Keywords

Normal

α-normal

β-normal

C-normal

Epinormal

Mildly normal

54D15

54B10

1 Introduction

2 C– α -Normality and C- β -Normality

Lemma 1.6 (Arhangel’skii and Ludwig, 2001)

Theorem 1.8 (Arhangel’skii and Ludwig, 2001)

Lemma 1.12 (Arhangel’skii and Ludwig, 2001)

3 C– α -Normality (C- β -Normality) and some other properties

4 Conclusion

Acknowledgments

Declaration of Competing Interest

References

Appendix A

Supplementary data

Appendix A

Supplementary data

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Some topological properties on C- $α$ -Normality and C- $β$ -Normality

2 C– $α$ -Normality and C- $β$ -Normality

3 C– $α$ -Normality (C- $β$ -Normality) and some other properties