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_yand 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 $y \in U$