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].(adsbygoogle = window.adsbygoogle || []).push({});

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Can't understand a detail in paracompactness->normality

Loading...

Similar Threads for Can't understand detail |
---|

I Understanding metric space definition through concrete examples |

I Help me understand a part of a proof |

**Physics Forums | Science Articles, Homework Help, Discussion**