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

Help with Hausdorff spaces

  1. Mar 31, 2008 #1
    Let X be a locally compact Hausdorff space, Y a subspace. Show that the quotient space X/Y is a Hausdorff space.

    My attempt at a solution:
    I don't have a solution. I cannot connect a Hausdorff space with a quotient space.

    Since X is compact Hausdorff x,y [tex]\in[/tex] X s.t. x and y can be seperated by neighborhoods if [tex]\exists[/tex] a neighborhood U of x and V of y s.t. U /\ V = [tex]\phi[/tex]. Now, somehow this implies that, for open sets U, the [tex]\bigcup[/tex] U [tex]\subset[/tex] X are disjoint.
  2. jcsd
  3. Mar 31, 2008 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Normally, a topological space is quotiented by an equivalence relation, not a subspace.
  4. Apr 1, 2008 #3


    User Avatar
    Science Advisor
    Homework Helper

    if you are collapsing the subspace Y to a point, maybe you want a closed subspace?

    then you would be asking whether any point not in that subspace can be separated from it by disjoint open neighborhoods.

    it still seems kind of odd. i guess i need more definitions. what does locally compact mean?

    in a hausdorff space, distinct points have disjoint open neighborhoods, so you want them to still be disjoint after collapsing Y? so you need disjoint nbhds that also miss Y? (and thats for points not themselves contained in Y.....
    Last edited: Apr 1, 2008
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook