Some topological properties on C- $\mathit{\alpha }$ -Normality and C- $\mathit{\beta }$ -Normality

A topological space $(Y,\tau )$ is called C- $\alpha$ -normal (C- $\beta$ -normal) if there exist a bijective function $g$ from $Y$ onto $\alpha$ -normal ( $\beta$ -normal) space $Z$ such that the restriction map $g_{{|}_B}$ from $B$ onto $g\left(B\right)$ is a homeomorphism for any compact subspace $B$ of $Y$ .

Any regular space is $\alpha$ -regular.

Let $(Y,\tau )$ 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 $\overline{F\cap W_2}=F$ (note that $\overline{F}=F$ since $F$ is closed), and $W_1\cap W_2=\varnothing$ , therefore $(Y,\tau )$ is $\alpha$ -regular space.

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

Any regular space is almost $\alpha$ -regular.

Any normal space is $\alpha$ -normal.

Lemma 1.6 (Arhangel’skii and Ludwig, 2001)

Any normal space is $\beta$ -normal.

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 $W_1\cap W_2=\varnothing$ . Hence $\overline{F_1\cap W_1}=F_1$ and $\overline{F_2\cap W_2}=F_2$ . It remains to prove $\overline{W_1}\cap \overline{W_2}=\varnothing$ . 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 ⊆ $\overline{V}$  ⊆ U. Now apply this to $F_1$ and set $W_1$  = V and $W_2$  = Y\ $\overline{V}$ .

Any C-normal space is C- $\alpha$ -normal (C- $\beta$ -normal).

Theorem 1.8 (Arhangel’skii and Ludwig, 2001)

Any $\alpha$ -normal extremally disconnected space is normal.

Let $Y$ be a $\alpha$ -normal extremally disconnected space. Pick two disjoint closed sets $F_1$ and $F_2$ subsets of $Y$ . Since $Y$ is $\alpha$ -normal, then there exist two disjoint open sets $W_1$ and $W_2$ subsets of $Y$ where $\overline{F_1\cap W_1}=F_1$ and $\overline{F_2\cap W_2}=F_2$ . Hence $F_1\subseteq \overline{W_1}$ and $F_2\subseteq \overline{W_2}$ . However, $\overline{W_1}$ ∩ $W_2$ = ∅ since $W_2$ is open and $W_1$ ∩ $W_2$ = ∅. Thus, $\overline{W_1}$ ∩ $\overline{W_2}$ = ∅ as well since $\overline{W_1}$ is open (by extremally disconnectedness) and $\overline{W_1}$ ∩ $W_2$ = ∅.

If $Y$ is $C$ - $\alpha$ -normal ( $C$ - $\beta$ -normal) such that the witness of $C$ - $\alpha$ -normal ( $C$ - $\beta$ -normal) is extremally disconnected, then $Y$ is $C$ -normal.

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

Let $Y$ be C- $\beta$ -normal. Then the codomain $Z$ witness of C- $\beta$ -normal is $\beta$ -normal. Let $F_1$ and $F_2$ be any disjoint closed subsets of $Z$ . Since $Z$ is $\beta$ -normal, there exist open subsets $W_1$ and $W_2$ of $Z$ where $\overline{W_1}\cap \overline{W_2}=\varnothing$ , $\overline{F_1\cap W_1}=F_1$ and $\overline{F_2\cap W_2}=F_2$ . So $\overline{W_1}$ , $\overline{W_2}$ 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 $F_1\subseteq \overline{W_1}\subseteq U_1$ and $F_2\subseteq \overline{W_2}\subseteq U_2$ . Hence $Z$ is normal.

Any $\alpha$ -normal space satisfying $T_1$ axiom is Hausdorff.

Let $Y$ be any $\alpha$ -normal $T_1$ -space. Let $y,z$ be any two distinct elements in $Y$ . Hence $\left\{y\right\}$ and $\left\{z\right\}$ are disjoint closed subsets of $Y$ , by $\alpha$ -normality, there exist two disjoint open subsets $G_1$ and $G_2$ of $Y$ where $\overline{\left\{y\right\}\cap G_1}=\left\{y\right\}$ and $\overline{\left\{z\right\}\cap G_2}=\left\{z\right\}$ which implies $y\in G_1$ and $z\in G_2$ . Therefore $Y$ is Hausdorff.

Lemma 1.12 (Arhangel’skii and Ludwig, 2001)

Any $\beta$ -normal space satisfying $T_1$ axiom is regular (hence Hausdorff).

Any $\beta$ -normal space satisfying $T_1$ axiom is almost $\alpha$ -regular.

Any $\beta$ -normal space satisfying $T_1$ axiom is $\alpha$ -regular.

Any $\alpha$ -normal space satisfying $T_1$ axiom is $\alpha$ -regular.

Any $\alpha$ -normal space satisfying $T_1$ axiom is almost $\alpha$ -regular.

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

Every submetrizable space is C- $\alpha$ -normal (C- $\beta$ -normal).

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

The Rational Sequence Topology $(\mathbb{R},\mathcal{R}S)$ (Steen and Seebach, 1995) is submetrizable being finer than the usual topology $(\mathbb{R},\mathcal{U})$ , so $(\mathbb{R},\mathcal{R}S)$ is C- $\alpha$ -normal (C- $\beta$ -normal).

1. The Half-Disc topological space (Steen and Seebach, 1995) is C- $\alpha$ -normal (C- $\beta$ -normal) because it is submetrizable by Theorem 1.18. but it is not $\alpha$ -normal nor $\beta$ -normal because it is first countable and Hausdorff but not regular, so by Proposition 1.17. the Half-Disc topological space is not $\alpha$ -normal space, hence not $\beta$ -normal. In general C- $\alpha$ -normality (C- $\beta$ -normality) do not imply $\alpha$ -normality ( $\beta$ -normality) even with Hausdorff or first countable properties.

2. The Deleted Tychonoff Plank (Steen and Seebach, 1995), it is C- $\alpha$ -normal (C- $\beta$ -normal) since it is locally compact by Theorem 2.7. but it is not $\alpha$ -normal nor $\beta$ -normal see (Arhangel’skii and Ludwig, 2001).

3. The Dieudonn $\stackrel{Ấ}{e}$ Plank (AlZahrani and Kalantan, 2017), in example 1.10 we proved that it is C-normal, hence it is C- $\alpha$ -normal (C- $\beta$ -normal) by Theorem 1.9. but it is not $\alpha$ -normal nor $\beta$ -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.

4. The Sorgenfrey line square $\mathcal{S}\times \mathcal{S}$ see (Steen and Seebach, 1995) is not normal, but it is submetrizable space being it is finer than the usual topology on $\mathbb{R}\times \mathbb{R}$ , so by Theorem 1.18. it is C- $\alpha$ -normal (C- $\beta$ -normal).

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

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

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

Let $\mathbb{R}$ with the countable complement topology $\mathcal{C}\mathcal{C}$ (Steen and Seebach, 1995). We know $(\mathbb{R},\mathcal{C}\mathcal{C})$ is $T_1$ and the only compact sets are finite, hence the compact subspaces are discrete. If we let $\mathcal{D}$ be the discrete topology on $\mathbb{R}$ , then obviously the identity function from $(\mathbb{R},\mathcal{C}\mathcal{C})$ onto $(\mathbb{R},D)$ is a witnessing of the C- $\alpha$ -normality (C- $\beta$ -normality) which is not continuous.

If $(Y,\tau )$ is a C- $\alpha$ -normal (C- $\beta$ -normal) and Fréchet space, then any function witnessing of C- $\alpha$ -normality (C- $\beta$ -normality) is continuous.

Let $Y$ be a Fréchet C- $\alpha$ -normal (C- $\beta$ -normal) space and $g:Y\to Z$ be a witness of the C- $\alpha$ -normality (C- $\beta$ -normality) of $Y$ . Let $A\subseteq Y$ and pick $z\in g\left(\overline{A}\right)$ . There is a unique $y\in Y$ where $g\left(y\right)=z,$ thus $y\in \overline{A}$ . since $Y$ is Fréchet, then there exists a sequence $\left(a_n\right)\subseteq A$ where $a_n\to y$ . As the subspace $K=\left\{y\right\}\cup \left\{a_n:n\in \mathbb{N}\right\}$ of $Y$ is compact, the induced map $g_{{|}_K}:K\to g\left(K\right)$ is a homeomorphism. Let $U\subseteq Z$ be any open neighborhood of $z$ . Then $U\cap g\left(K\right)$ is an open neighborhood of $z$ in the subspace $g\left(K\right)$ . Since $g_{{|}_K}$ is a homeomorphism, then $g^{-1}(U\cap g\left(K\right))=g^{-1}\left(U\right)\cap K$ is an open neighborhood of y in K, then there exists $m\in \mathbb{N}$ where $a_n\in g^{-1}(U\cap g\left(K\right))$ $\forall n\ge m,$ hence $g\left(a_n\right)\in \left(U\cap g\left(K\right)\right)\forall n\ge m,$ then $U\cap g\left(A\right)\ne \varnothing .$ Hence $z\in \overline{g\left(A\right)}$ and $g\left(\overline{A}\right)\subseteq \overline{g\left(A\right)}.$ Thus $g$ is continuous.

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

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

If $Y$ is F-compact, then $Y$ is $C$ - $\alpha$ -normal ( $C$ - $\beta$ -normal).

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.

Consider $(\mathbb{R},\mathcal{C}\mathcal{C})$ , where $\mathcal{C}\mathcal{C}$ is the countable complement topology (Steen and Seebach, 1995). We know $(\mathbb{R},\mathcal{C}\mathcal{C})$ is $T_1$ and the only compact sets are finite, therefore, by Theorem 1.25. $(\mathbb{R},\mathcal{C}\mathcal{C})$ is $C$ - $\alpha$ -normal ( $C$ - $\beta$ -normal). This a fourth example of $C$ - $\alpha$ -normal ( $C$ - $\beta$ -normal) but not $\alpha$ -normal (nor $\beta$ -normal).

C- $\alpha$ -normality (C- $\beta$ -normality) is a topological property.

Let $Y$ be a C- $\alpha$ -normal (C- $\beta$ -normal) space and let $Y\cong W$ . Let $Z$ be a $\alpha$ -normal ( $\beta$ -normal) space and let $g:Y\to Z$ be a bijective function where the restriction map $g_{{|}_B}$ from $B$ onto $g\left(B\right)$ 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.

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

If $Y$ is C- $\alpha$ -normal (C- $\beta$ -normal) space and the witness of the C- $\alpha$ -normality (C- $\beta$ -normality) of $Y$ is $T_1$ , then $Y$ is C- $\alpha$ -regular.

If $Y$ is C- $\beta$ -normal space and the codomain witness of the C- $\beta$ -normality of $Y$ is $T_1$ , then $Y$ is C-regular.

If $Y$ is a C- $\alpha$ -normal (C- $\beta$ -normal) Fréchet space and the witness of the C- $\alpha$ -normality (C- $\beta$ -normality) is $T_1$ , then $Y$ is $T_2$ .

Let $\mathrm{Y}$ is a C- $\mathrm{\alpha }$ -normal (C- $\mathrm{\beta }$ -normal) Fréchet space, then there exist $\mathrm{\alpha }$ -normal ( $\mathrm{\beta }$ -normal) space $\mathrm{Z}$ (witness of the C- $\mathrm{\alpha }$ -normality (C- $\mathrm{\beta }$ -normality)) and a bijective function $g:\mathrm{Y}\to \mathrm{Z}$ such that the restriction map $g_{{|}_B}$ from $B$ onto $g\left(B\right)$ 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 $a\ne b$ , then $g\left(a\right)\ne g\left(b\right)$ , $g\left(a\right),g\left(b\right)\in Z$ . Since $Z$ is $\mathrm{\alpha }$ -normal ( $\mathrm{\beta }$ -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 $g\left(a\right)\in W_1,g\left(b\right)\in W_2$ and $W_1\cap W_2=\varnothing$ . Since $W_1,W_2$ are open sets in $Z$ and $g$ is continuous, then $g^{-1}\left(W_1\right)$ and $g^{-1}\left(W_2\right)$ are open sets in $Y$ , $a\in {g^{-1}(W}_1),b\in {g^{-1}(W}_2)$ and ${g^{-1}(W}_1)\cap {g^{-1}(W}_2)={g^{-1}(W}_1\cap W_2)=\varnothing$ . Hence $Y$ is $T_2$ .

Any C-regular Fréchet Lindelof space is C- $\alpha$ -normal (C- $\beta$ -normal).

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- $\alpha$ -normal (C- $\beta$ -normal).

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

Every locally compact space is C- $\alpha$ -normal (C- $\beta$ -normal).

Let $Y$ be locally compact space. By (Engelking, 1977), 3.3.D], there exists $T_2$ compact space $Z$ and hence $\alpha$ -normal ( $\beta$ -normal), and a continuous bijective function $g:Y\to Z$ . We have $g_{{|}_K}$ from $K$ onto $g\left(K\right)$ 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 $T_2$ .

Consider ${\omega }_1$ , the first uncountable ordinal, we consider ${\omega }_1$ as an open subspace of its successor $({\omega }_1+1)$ , which is compact and hence is locally compact [14, Example 43]. Thus, ${\omega }_1$ is locally compact as an open subspace of a locally compact space, see (Engelking, 1977),3.3.8]. Then by Theorem 2.7. ${\omega }_1$ is C- $\alpha$ -normal (C- $\beta$ -normal).

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

Every epinormal space is C- $\alpha$ -normal (C- $\beta$ -normal).

Every epi- $\alpha$ -normal (epi- $\beta$ -normal)space is C- $\alpha$ -normal(C- $\beta$ -normal).

Any C- $\alpha$ -normal (C- $\beta$ -normal) Fréchet space is epi- $\alpha$ -normal (epi- $\beta$ -normal).

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

Any lower compact space is C- $\alpha$ -normal (C- $\beta$ -normal).

Let $(Y,\tau )$ is lower compact, then $(Y,{\tau }^{\prime })$ is $T_2$ -compact, hence normal and the identity function $i{d}_Y:(Y,\tau )\to (Y,{\tau }^{\prime })$ is a continuous and bijective. If we take $B$ any compact subspace of $(Y,\tau )$ , then ${i{d}_Y}_{{|}_B}$ is a homeomorphism by (Engelking, 1977);3.1.13].

If $(Y,\mathrm{\tau })$ is C- $\alpha$ -normal compact Fréchet space and the witness of the C- $\alpha$ -normality is $T_1$ , then $(Y,\tau )$ is lower compact.

Pick $\mathrm{\alpha }$ -normal space $(\mathrm{Z},{\mathrm{\tau }}^{\ast })$ and a bijective function $g:(Y,\tau )\to (Z,{\tau }^{\ast })$ such that $g_{{|}_B}:B\to g\left(B\right)$ is a homeomorphism for any compact subspace $B\subseteq Y$ . Since $Y$ is Fréchet, then $g$ is continuous. Hence $(\mathrm{Z},{\mathrm{\tau }}^{\ast })$ is compact. Since $(\mathrm{Z},{\mathrm{\tau }}^{\ast })$ is $T_1$ $\mathrm{\alpha }$ -normal space, then by Lemma 1.11. it is Hausdorff. Hence $(\mathrm{Z},{\mathrm{\tau }}^{\ast })$ is $T_2$ compact. Define a topology ${\mathrm{\tau }}^{\prime }$ on $Y$ as follows ${\mathrm{\tau }}^{\prime }=\left\{g^{-1}\left(V\right):V\in {\tau }^{\ast }\right\}.$ Then ${\mathrm{\tau }}^{\prime }$ is coarser than $\tau$ and $g:(Y,{\tau }^{\prime })\to (Z,{\tau }^{\ast })$ is a bijection continuous function. Let any $U\in {\tau }^{\prime }$ , then $U$ is of the form $g^{-1}\left(V\right)$ for some $V\in {\tau }^{\ast }$ . Hence $g\left(U\right)=g\left(g^{-1}\left(V\right)\right)=V$ . Thus $g$ is open. Hence $g$ is a homeomorphism. So $(Y,{\tau }^{\prime })$ is $T_2$ compact. Therefore $(Y,\tau )$ is lower compact.

If $(Y,\mathrm{\tau })$ is C- $\beta$ -normal compact Fréchet space and the witness of the C- $\beta$ -normality is $T_1$ , then $(Y,\tau )$ is lower compact.