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Can't understand a detail in paracompactness->normality

  1. May 22, 2015 #1
    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].
     
    Last edited: May 22, 2015
  2. jcsd
  3. May 22, 2015 #2

    wabbit

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    Re-read the definition of ## U ## - is ## K ## one of those sets ## W \in B##?
     
  4. May 22, 2015 #3
    I think get it now. Willard states that a topological space [itex]X[/itex] is paracomact iff any cover of [itex]X[/itex] has an open locally finite refinement. It does not necessarilly implies that it covers [itex]X[/itex]. Munkres however states that this refinement does cover [itex]X[/itex], so [itex]U[/itex] should cover [itex]K[/itex].
     
  5. May 22, 2015 #4

    wabbit

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    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##
     
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