Countable paracompactness

Subjects: Topology
Links: Paracompacteness, Countable compactness, Collectionwise Normal Spaces, Metacompactness

Def: A topological space $X$ is countably paracompact if every countable open cover of $X$ has a locally finite open refinement.

Obs: Every paracompact space is countably paracompact. Every countable compact space is coumtably paracompact.

Th: The following properties are equivalent for a topological space $X$.

• $X$ is countably paracompact.
• Every countable open cover of $\{U_n\mid n<\omega \}$ has a refinement $\{V_n\mid n<\omega \}$ with $V_n\subseteq U_n$ for each $n<\omega$.
• For every increasing sequence of open subsets $\{U_n\mid n<\omega \}$ of $X$ satisfying $X=\bigcup _{n<\omega }{U}_n$ there exists a sequence of $\{F_n\mid n<\omega \}$ of closed subsets of $X$ such that $F_n\subseteq W_n$ for each $n<\omega$ and $\bigcup _{n<\omega }\text{int}(F_n)=X$.
• For every decreasing sequence $\{F_n\mid n<\omega \}$ of closed subsets such that $\bigcap _{n<\omega }{F}_n=\varnothing$, there is a sequence $\{U_n\mid n<\omega \}$ of open sets such that $U_n\supseteq F_n$, and $\bigcap _{n<\omega }\text{cl}(U_n)=\varnothing$.

Th: The following properties of an normal space are equivalent.

• $X$ is normal countably paracompact.
• Every countable open cover of $\{U_n\mid n<\omega \}$ has a refinement $\{V_n\mid n<\omega \}$ with $\text{cl}(V_n)\subseteq U_n$ for each $n<\omega$.
• For every increasing sequence of open subsets $\{U_n\mid n<\omega \}$ of $X$ satisfying $X=\bigcup _{n<\omega }{U}_n$ there exists a sequence of $\{F_n\mid n<\omega \}$ of closed subsets of $X$ such that $F_n\subseteq W_n$ for each $n<\omega$ and covers $X$.

Prop: Every countable cover $\{U_n\mid n<\omega \}$ of a topological space $X$, where each $U_n$ is a cozero set, has a countable star-finite refinement consisting of cozero sets.

Cor: Every perfectly normal space is countably paracompact.

Th: For a normal space $X$ the following conditions are equivalent.

• $X$ is countably paracompact.
• $X$ is countably metacompact.
• $X$ is countably strongly paracompact (every countable cover of the space $X$ has a star-finite open refinement).

Lemma: The Cartesian product $X\times Y$ of a countably paracompact normal space $X$ and a compact second-countable space $Y$ is normal.

Th: A topological space $X$ is countably paracompact normal space iff the Cartesian product $X\times [0,1]$ is normal.

Prop: The following are equivalent for a normal space $X$.

• $X$ is collectionwise normal and countably paracompact.
• Every locally finite closed collection $\{F_{\alpha }\mid \alpha <\kappa \}$ in $X$, there is a locally finite open collection $\{U_{\alpha }\mid \alpha <\kappa \}$ such that $F_{\alpha }\subseteq U_{\alpha }$ for all $\alpha <\kappa$.

We can generalise further the idea of contable paracompactness.

Def: Let $\kappa$ be an infinite cardinal and $X$ be a topological space. If every open cover $\mathcal{U}$ of $X$ with $|\mathcal{U}|\le \kappa$ has a locally finite open refinement, then $X$ is called a $\kappa$-paracompact.

Obs: We say that $X$ is paracompact iff it is $\kappa$-paracompact for each $\kappa$.