Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook