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

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.

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 $f(N)\subset\{1\}$.

I didn't show the proof of the claim.

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.