# Countable open covering of normal space

I read the following claim in Z.Frolik's article "A generalization of realcompact spaces" on page 135.

Two subset $$M$$ and $$N$$ of a space $$X$$ are called completely seperated if there exists a real valued continuous function $$f$$ on $$X$$ with $$f(M)\subset \{0\}$$ and $$f(N)\subset\{1\}$$.

Claim: Let $$X$$ be a normal space. For every countable open covering $$\mathfrak{U}$$ of $$X$$, there exists a countable open covering $$\mathfrak{B}$$ of $$X$$ such that for every $$B$$ in $$\mathfrak{B}$$ there exists an $$A$$ in $$\mathfrak{U}$$ such that $$B$$ and $$X-A$$ are completely seperated.

I didn't show the proof of the claim.

• What is the question? Jan 2, 2021 at 20:39
• The existence of $\mathfrak{B}$ Jan 2, 2021 at 20:50

The claim is false it would imply that normal spaces are countably paracompact and hence that normality of $$X$$ would imply normality of $$X\times[0,1]$$. The latter is not the case, see Mary Ellen Rudin, A normal space $$X$$ for which $$X\times I$$ is not normal, Fundamenta Mathematicae, 73 (1971/72), 179-186.
To show that the property in the claim implies countable paracompactness we use Theorem 5.2.1 in Engelking's General Topology. Let $$\{U_n:n\in\omega\}$$ be an increasing open cover; we need to find open $$O_n$$ such that $$\operatorname{cl}O_n\subseteq U_n$$ for all $$n$$ and $$\bigcup_nO_n=X$$. Take an open cover $$\{V_m:m\in\omega\}$$ as in the claim; hence for every $$m$$ an $$n$$ such that $$\operatorname{cl}V_m\subseteq U_n$$. Now define $$O_n=\bigcup\{V_m:m\le n$$ and $$\operatorname{cl}V_m\subseteq U_n\}$$; then $$\operatorname{cl}O_n\subseteq U_n$$ for all $$n$$ and the $$O_n$$ form a cover (if $$x\in V_m$$ and $$\operatorname{cl}V_m\subseteq U_n$$ then $$x\in O_n$$).