# Decomposition, Mapping, and Sum Theorems of ω-Paracompact Topological Spaces

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2**

**Proposition**

**1**

- (a)
- Every countably paracompact topological space is countably ω-paracompact but not conversely;
- (b)
- Every countably ω-paracompact topological space is countably metacompact but not conversely;
- (c)
- Every ω-paracompact topological space is countably ω-paracompact but not conversely.

**Proposition**

**2**

- (a)
- $\left(X,\tau \right)$ is countably paracompact;
- (b)
- $\left(X,\tau \right)$ is countably metacompact.

**Proposition**

**3.**

- (a)
- $\left(X,\tau \right)$ is countably paracompact;
- (b)
- $\left(X,\tau \right)$ is countably ω-paracompact;
- (c)
- $\left(X,\tau \right)$ is countably metacompact.

**Proof.**

**Proposition**

**4**

**Proposition**

**5**

- (a)
- $\left(X,\tau \right)$is countably ω-paracompact;
- (b)
- For every countable open cover $\left\{{A}_{n}:n\in \mathbb{N}\right\}$ of X, there exists an ω-locally finite open cover $\left\{{B}_{n}:n\in \mathbb{N}\right\}$ of X such that for all $n\in \mathbb{N}$, ${A}_{n}\subseteq {B}_{n}$.

**Proposition**

**6**

**Proposition**

**7**

**Definition**

**3**

**Proposition**

**8**

**Definition**

**4**

**P**be any topological property. We say that the locally finite sum theorem holds for

**P**if the following is satisfied:

**P**, then $\left(X,\tau \right)$ possesses the property

**P**.

**Proposition**

**9**

**P**be a property satisfying the following:

- (a)
- The disjoint sum of topological spaces possessing the property
**P**possesses**P**; - (b)
**P**is preserved under closed continuous mappings with finite fibers.

**P**.

## 3. Results

**Definition**

**5.**

**Definition**

**6.**

- (a)
- σ-ω-paracompact if every open cover has an open σ-ω-locally finite refinement;
- (b)
- Feebly ω-paracompact if every open cover of X has an ω-locally finite refinement.

**Proposition**

**11.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Claim**

**1.**

**Proof**

**of**

**Claim**

**1.**

**Claim**

**2.**

- (i)
- $\mathcal{E}$covers X;
- (ii)
- $\mathcal{E}$refines$\mathcal{A}$;
- (iii)
- $\mathcal{E}$is $\omega $-locally finite.

**Proof**

**of**

**Claim**

**2.**

- (i)
- Let $x\in X$. Let ${n}_{x}$ be the smallest natural number such that $x\in {C}_{{n}_{x}}$. Since ${C}_{{n}_{x}}=\cup \left\{B:B\in {\mathcal{B}}_{{n}_{x}}\right\}$, then there exists ${B}_{x}\in {\mathcal{B}}_{{n}_{x}}$ such that $x\in {B}_{x}$. Thus, $x\in {D}_{{n}_{x}}\cap {B}_{x}\in {\mathcal{E}}_{{n}_{x}}\subseteq \mathcal{E}$.
- (ii)
- Let $E\in \mathcal{E}$. Then, there exists ${n}_{0}\in \mathbb{N}$ and ${B}_{0}\in {\mathcal{B}}_{{n}_{0}}$ such that $E={B}_{0}\cap {D}_{{n}_{0}}$. Since $\mathcal{B}$ refines $\mathcal{A}$ and ${B}_{0}\in {\mathcal{B}}_{n}$, then there exists ${A}_{0}\in \mathcal{A}$ such that ${B}_{0}\subseteq {A}_{0}$, and thus $E\subseteq {A}_{0}$.
- (iii)
- Let $x\in X$. By Claim 1, there exists ${O}_{x}\in \tau $ such that $x\in {O}_{x}$ and ${O}_{x}$ intersects at most ${D}_{{n}_{1}},{D}_{{n}_{2}},\dots ,{D}_{{n}_{k}}$ of $\mathcal{D}$. For each $i=1,2,\dots ,k$, we have ${\mathcal{B}}_{{n}_{i}}$ is $\omega $-locally finite and so ${\mathcal{E}}_{{n}_{i}}$ is $\omega $-locally finite. Thus, for each $i=1,2,\dots ,k$, there is an $\omega $-open set ${O}_{i}$ such that $x\in {O}_{i}$ and ${O}_{i}$ intersects at most finitely many elements of ${\mathcal{E}}_{{n}_{i}}$. Let $O={O}_{x}\cap \left({\cap}_{i=1}^{k}{O}_{i}\right)$. Then, O is $\omega $-open, $x\in O$, and O intersects at most finitely many elements of $\mathcal{E}$. □

**Example**

**1.**

**Proposition**

**12.**

**Proof.**

**Claim**

**3.**

- (i)
- $\mathcal{B}$refines$\mathcal{A}$.
- (ii)
- $\mathcal{B}$is $\omega $-locally finite.

**Proof**

**of**

**Claim**

**3.**

- (i)
- Let $B\in \mathcal{B}$, say $B=\left\{y\right\}$ for some $y\in X$. Since $\mathcal{A}$ is a cover of X, then there exists $A\in \mathcal{A}$ such that $y\in A$. Thus, we have $A\in \mathcal{A}$ with $B\subseteq A$. It follows that $\mathcal{B}$ refines $\mathcal{A}$.
- (ii)
- Let $y\in X$. Let $O=\left\{y\right\}$. Since $\left(X,\tau \right)$ is locally countable, then O is $\omega $-open. Thus, we have $y\in O$, O is $\omega $-open, and $\left\{B:O\cap B\ne \varnothing \right\}=\left\{\left\{y\right\}\right\}$ which is finite. It follows that $\mathcal{B}$ is $\omega $-locally finite. □

**Example**

**2.**

**Claim**

**4.**

- (i)
- $\mathcal{B}-\left\{\varnothing ,\left\{p\right\}\right\}=\mathcal{A}$.
- (ii)
- $\mathcal{B}\phantom{\rule{4pt}{0ex}}$is not $\sigma $-$\omega $-locally finite.

**Proof**

**of**

**Claim**

**4.**

- (i)
- Let $B\in \mathcal{B}-\left\{\varnothing ,\left\{p\right\}\right\}$. Since $B\in \tau $, then there is $x\in X-\left\{p\right\}$ such that $\left\{p,x\right\}\subseteq B$. Since $\mathcal{B}$ refines $\mathcal{A}$, then there is $A\in $$\mathcal{A}$ such that $B\subseteq A$. Therefore, $A=\left\{p,x\right\}=B$, and hence $B\in \mathcal{A}$. This shows that $\mathcal{B}-\left\{\varnothing ,\left\{p\right\}\right\}\subseteq \mathcal{A}$. To see that $\mathcal{A}\subseteq \mathcal{B}-\left\{\varnothing ,\left\{p\right\}\right\}$, let $A\in \mathcal{A}$. Then, there exists $x\in X-\left\{p\right\}$ such that $\left\{p,x\right\}=A$. Since $\mathcal{B}$ is a cover of X, then there is $B\in \mathcal{B}$ such that $x\in B$. Since $B\in \tau $, then $\left\{p,x\right\}\subseteq B$. Since $\mathcal{B}$ refines $\mathcal{A}$, then there is ${A}_{0}\in $$\mathcal{A}$ such that $B\subseteq {A}_{0}$. Therefore, $A=\left\{p,x\right\}=B={A}_{0}$ and hence $A\in \mathcal{B}-\left\{\varnothing ,\left\{p\right\}\right\}$.
- (ii)
- Suppose to the contrary that $\mathcal{B}$ is $\sigma $-$\omega $-locally finite, then $\mathcal{B}=$${\cup}_{n=1}^{\infty}{\mathcal{B}}_{n}$ where ${\mathcal{B}}_{n}$ is $\omega $-locally finite for all $n\in \mathbb{N}$. Since X is uncountable, then there are ${n}_{0}\in \mathbb{N}$ and $Y\subseteq X$ such that Y is uncountable and ${\mathcal{B}}_{{n}_{0}}-\left\{\varnothing ,\left\{p\right\}\right\}=\left\{\left\{p,y\right\}:y\in Y\right\}$. Since ${\mathcal{B}}_{{n}_{0}}$ is $\omega $-locally finite, then there is an $\omega $-open set O in X such that $p\in O$ and $\left\{B:O\cap B\ne \varnothing \right\}$ is finite which is a contradiction. □

**Example**

**3.**

**Claim**

**5.**

- (i)
- $\mathcal{B}\phantom{\rule{4pt}{0ex}}$refines$\mathcal{A}$.
- (ii)
- $\mathcal{B}\phantom{\rule{4pt}{0ex}}$is $\sigma $-$\omega $-locally finite.

**Proof**

**of**

**Claim**

**5.**

- (i)
- Let $B\in \mathcal{B}$. If $B=\left\{p\right\}$, then choose $A\in \mathcal{A}$ such that $p\in A$, and thus we have $A\in \mathcal{A}$ with $B\subseteq A$. If $B=\left\{p,x\right\}$ for some $x\in X-\left\{p\right\}$, then choose $A\in \mathcal{A}$ such that $x\in A$, and thus we have $A\in \mathcal{A}$ with $B=\left\{p,x\right\}\subseteq A$. It follows that $\mathcal{B}$ refines $\mathcal{A}$.
- (ii)
- Since X is a countable infinite set, then we can write $X=\left\{{x}_{n}:n\in \mathbb{N}\right\}$ with ${x}_{1}=p$ and ${x}_{n}\ne {x}_{m}$ for $n\ne m$. Let ${\mathcal{B}}_{1}=\left\{\left\{p\right\}\right\}$, and for every $n\in \mathbb{N}-\left\{1\right\}$ let ${\mathcal{B}}_{n}=\left\{\left\{p,{x}_{n}\right\}\right\}$. Then, $\mathcal{B}=$${\cup}_{i=1}^{\infty}{\mathcal{B}}_{n}$. Since ${\mathcal{B}}_{n}$ is clearly $\omega $-locally finite for all $n\in \mathbb{N}$, then $\mathcal{B}$ is $\sigma $-$\omega $-locally finite. □

**Claim**

**6.**

- (i)
- $\mathcal{B}-\left\{\varnothing ,\left\{p\right\}\right\}=\mathcal{A}$;
- (ii)
- $\mathcal{B}\phantom{\rule{4pt}{0ex}}$is not $\omega $-locally finite.

**Proof**

**of**

**Claim**

**6.**

- (i)
- Let $B\in \mathcal{B}-\left\{\varnothing ,\left\{p\right\}\right\}$. Since $B\in \tau $, then there exists $x\in X-\left\{p\right\}$ such that $\left\{p,x\right\}\subseteq B$. Since $\mathcal{B}$ refines $\mathcal{A}$, then there exists $A\in $$\mathcal{A}$ such that $B\subseteq A$. Therefore, $A=\left\{p,x\right\}=B$ and so $B\in \mathcal{A}$. This shows that $\mathcal{B}-\left\{\varnothing ,\left\{p\right\}\right\}\subseteq \mathcal{A}$. To see that $\mathcal{A}\subseteq \mathcal{B}-\left\{\varnothing ,\left\{p\right\}\right\}$, let $A\in \mathcal{A}$. Then, there exists $x\in X-\left\{p\right\}$ such that $\left\{p,x\right\}=A$. Since $\mathcal{B}$ is a cover of X, then there exists $B\in \mathcal{B}$ such that $x\in B$. Since $B\in \tau $, then $\left\{p,x\right\}\subseteq B$. Since $\mathcal{B}$ refines $\mathcal{A}$, then there exists ${A}_{0}\in $$\mathcal{A}$ such that $B\subseteq {A}_{0}$. Therefore, $A=\left\{p,x\right\}=B={A}_{0}$ and so $A\in \mathcal{B}-\left\{\varnothing ,\left\{p\right\}\right\}$.
- (ii)
- Suppose to the contrary that $\mathcal{B}$ is $\omega $-locally finite. Since $\mathcal{B}$ is $\omega $-locally finite, then there is an $\omega $-open set O in X such that $p\in O$ and $\left\{B:O\cap B\ne \varnothing \right\}$ is finite which is a contradiction. □

**Question**

**1.**

**Question**

**2.**

**Question**

**3.**

**Proof.**

**Claim**

**7.**

**Proof**

**of**

**Claim**

**7.**

**Question**

**4.**

**Theorem**

**2.**

**Proof.**

**Claim**

**8.**

**Proof**

**of**

**Claim**

**8.**

**Claim**

**9.**

**Proof**

**of**

**Claim**

**9.**

**Theorem**

**3.**

- (a)
- $\left(X,\tau \right)$is ω-paracompact;
- (b)
- $\left(X,\tau \right)$σ-ω-paracompact and countably ω-paracompact.

**Proof.**

- (a)
- ⟹ (b) follows from Propositions 1 (c) and 11.
- (b)
- ⟹ (a) follows from Theorem 2. □

**Lemma**

**1.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Claim**

**10.**

- (i)
- $\mathcal{E}$covers X;
- (ii)
- $\mathcal{E}$refines$\mathcal{A}$;
- (iii)
- Each ${\mathcal{E}}_{n}$ is $\omega $-locally finite;

**Proof**

**of**

**Claim**

**10.**

- (i)
- Let $x\in X$. Since $\mathcal{D}$ covers X, there exist ${n}_{x}\in \mathbb{N}$ and ${C}_{x}\in {\mathcal{C}}_{{n}_{x}}$ such that $x\in {f}^{-1}\left({C}_{x}\right)\subseteq {f}^{-1}\left({B}_{y\left({C}_{x}\right)}\right)\subseteq \cup \left\{A:A\in {\mathcal{A}}_{y\left({C}_{x}\right)}\right\}$. Choose ${A}_{x}\in {\mathcal{A}}_{y\left({C}_{x}\right)}$ such that $x\in $${A}_{x}$. Thus, we have $x\in {f}^{-1}\left({C}_{x}\right)\cap {A}_{x}$ where ${f}^{-1}\left({C}_{x}\right)\cap {A}_{x}\in \mathcal{E}$. It follows that $\mathcal{E}$ covers X.
- (ii)
- Obvious.
- (iii)
- Let $x\in X$. Since ${\mathcal{D}}_{n}$ is $\omega $-locally finite, then there exists an $\omega $-open set ${O}_{x}$ with $x\in {O}_{x}$, and there exists a finite subcollection ${\mathcal{C}}_{x}\subseteq {\mathcal{C}}_{n}$ such that for all $C\in \mathcal{C}-{\mathcal{C}}_{x},{O}_{x}\cap {f}^{-1}\left(C\right)=\varnothing $. It follows that ${O}_{x}$ meets at most the finite subcollection $\left\{{f}^{-1}\left(C\right)\cap A:A\in {\mathcal{A}}_{y\left(C\right)},C\in {\mathcal{C}}_{x}\right\}$ of ${\mathcal{E}}_{n}$. □

**Corollary**

**1**

**Proof.**

**Theorem**

**5.**

**Proof.**

- (1)
- ${H}_{\alpha ,n}\subseteq {f}^{-1}\left({A}_{\alpha}\right)\phantom{\rule{4.pt}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}\alpha \in \Delta \phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}n\in \mathbb{N}$.
- (2)
- $f\left({H}_{\alpha ,n}\right)\cap f\left({E}_{\alpha ,n-1}\right)=\varnothing \phantom{\rule{4.pt}{0ex}}\mathrm{where}\phantom{\rule{4.pt}{0ex}}{E}_{\alpha ,n-1}=X-{\cup}_{\beta \ge \alpha}{H}_{\beta ,n-1}\phantom{\rule{4.pt}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}n>1$.

**Claim**

**11.**

- (3)
- ${E}_{{\alpha}_{\circ},n}=X-{\cup}_{\alpha \ge {\alpha}_{0}}{H}_{\alpha ,n}\subseteq {\cup}_{\alpha <{\alpha}_{0}}\left(X-{\cup}_{\beta >\alpha}{H}_{\beta ,n}\right)$.

**Proof**

**of**

**Claim**

**11.**

- (4)
- $X={\cup}_{\alpha \in \Delta}\left(X-{\cup}_{\beta >\alpha}{H}_{\beta ,n}\right).$

**Claim**

**12.**

- (a)
- ${\cup}_{i=1}^{\infty}{\mathcal{V}}_{i}$covers Y.
- (b)
- ${\cup}_{i=1}^{\infty}{\mathcal{V}}_{i}$refines$\left\{{A}_{\alpha}:\alpha \in \Delta \right\}.$

**Proof**

**of**

**Claim**

**12.**

- (a)
- Let $y\in Y$. By (4), the smallest element $\alpha $ in $\Delta $ such that $y\in f\left(X-{\cup}_{\beta >\alpha}{H}_{\beta ,n}\right)$ for some $n\in \mathbb{N}$ exists, denote it by $\alpha \left(y\right)$ and take an integer $n\left(y\right)$ such that $y\in f\left(X-{\cup}_{\beta >\alpha \left(y\right)}{H}_{\beta ,n\left(y\right)-1}\right)$. Now, for $\alpha >\alpha \left(y\right)$:$${\cup}_{\beta \ge \alpha}{H}_{\beta ,n\left(y\right)-1}\subseteq {\cup}_{\beta >\alpha \left(y\right)}{H}_{\beta ,n\left(y\right)-1},\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\mathrm{so}\phantom{\rule{4.pt}{0ex}}X-{\cup}_{\beta >\alpha \left(y\right)}{H}_{\beta ,n\left(y\right)-1}\subseteq X-{\cup}_{\beta \ge \alpha}{H}_{\beta ,n\left(y\right)-1}.$$

- (5)
- $i.e.,\phantom{\rule{4pt}{0ex}}{f}^{-1}\left(y\right)\cap \left({\cup}_{\alpha >\alpha \left(y\right)}{H}_{\alpha ,n\left(y\right)}\right)=\varnothing .$

- (6)
- ${f}^{-1}\left(y\right)\subseteq {\cup}_{\alpha \ge \alpha \left(y\right)}{H}_{\alpha ,n\left(y\right)}.$

- (b)
- Since for all $\alpha \in \Delta $ and $n\in \mathbb{N}$, ${H}_{\alpha ,n}\subseteq {f}^{-1}\left({A}_{\alpha}\right)$ and ${f}^{-1}\left({V}_{\alpha ,n}\right)\subseteq {H}_{\alpha ,n}$, then we have ${f}^{-1}\left({V}_{\alpha ,n}\right)\subseteq {f}^{-1}\left({A}_{\alpha}\right)$, and hence ${V}_{\alpha ,n}\subseteq {A}_{\alpha}$. □

**Example**

**4.**

**Proposition**

**13.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Corollary**

**2.**

## 4. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Al Ghour, S. Decomposition, Mapping, and Sum Theorems of *ω*-Paracompact Topological Spaces. *Axioms* **2021**, *10*, 339.
https://doi.org/10.3390/axioms10040339

**AMA Style**

Al Ghour S. Decomposition, Mapping, and Sum Theorems of *ω*-Paracompact Topological Spaces. *Axioms*. 2021; 10(4):339.
https://doi.org/10.3390/axioms10040339

**Chicago/Turabian Style**

Al Ghour, Samer. 2021. "Decomposition, Mapping, and Sum Theorems of *ω*-Paracompact Topological Spaces" *Axioms* 10, no. 4: 339.
https://doi.org/10.3390/axioms10040339