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One-point compactification

  1. Apr 28, 2012 #1
    Hi guys, I am confused about the definition of compactification of a topological space.

    Suppose (X,τx) is a topological space. Define Y=X[itex]\cup[/itex]{p} and a new topology τY such that U[itex]\subseteq[/itex]Y is open if
    (1) p [itex]\notin[/itex] U and U[itex]\in[/itex] [itex]\tau[/itex]X or
    (2) p [itex]\in[/itex] U and X-U is a compact closed subset of X.

    To prove that (Y,τY) is compact, it seems to require X-U in (2) to be compact under τY and not τX. That is if p[itex]\in[/itex] U, then any open covering {Vi}[itex]\subseteq[/itex]τY of Uc=X-U, has a finite subcover. But we don't even know what our open sets {Vi} are in the first place.

    Any help would be appreciated.
  2. jcsd
  3. Apr 28, 2012 #2


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    Staff Emeritus
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    The induced topology of X as a subspace of Y is the same as the original topology of X (this is not obvious, but not hard to prove, just some definition chasing), so if we have
    [tex] A\subset X \subset Y[/tex]

    A is compact in the topology of Y if and only if it is compact in the topology of X
  4. Apr 28, 2012 #3
    Ah, I see. Thanks very much.
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