Can't understand a detail in paracompactness->normality

[kostas230]

I seem to have stuck an obvious(?) detail in the proof of this theorem. We first show that a Hausdorff paracompact space is regular. Let $X$ be a Hausdorff paracompact space, $K$ be a closed subset of $X$, and $x\in X-K$. Since $X$ is Hausdorff there exists an open cover $\{ V_y: \; y\in K \}$ such that $y\in V_y$and $x \in X-\overline{V}_y$. Let $B$ be a locally finite refinement of the collection $\{ V_y: \; y\in K \}\cup\{X-K\}$ and $U = \cup \{W\in B: \; K\cap W\neq \emptyset\}$. What I cannot understand is why $U$ contains $K$.

 

[wabbit]

Re-read the definition of $U$ - is $K$ one of those sets $W \in B$?

 

[kostas230]

I think get it now. Willard states that a topological space $X$ is paracomact iff any cover of $X$ has an open locally finite refinement. It does not necessarilly implies that it covers $X$. Munkres however states that this refinement does cover $X$, so $U$ should cover $K$.

 

[wabbit]

Yeah forget what I said, that was silly. The point is that any $y\in K$ is in some $W\in B$ since $B$ is a cover. Obviously for that $W, y\in W\cap K$ so $W\cap K \neq \emptyset$, hence $y \in U$