- #1

kostas230

- 96

- 3

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 [itex]X [/itex] be a Hausdorff paracompact space, [itex]K[/itex] be a closed subset of [itex]X [/itex], and [itex]x\in X-K [/itex]. Since [itex]X [/itex] is Hausdorff there exists an open cover [itex]\{ V_y: \; y\in K \}[/itex] such that [itex]y\in V_y[/itex]and [itex]x \in X-\overline{V}_y[/itex]. Let [itex]B[/itex] be a locally finite refinement of the collection [itex]\{ V_y: \; y\in K \}\cup\{X-K\}[/itex] and [itex]U = \cup \{W\in B: \; K\cap W\neq \emptyset\}[/itex]. What I cannot understand is why [itex]U[/itex] contains [itex]K[/itex].

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