1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Singleton sets closed in T_1 and Hausdorff spaces

  1. Apr 17, 2010 #1
    If [tex](X,\tau)[/tex] is either a [tex]T_1[/tex] space or Hausdorff space then for any [tex]x \in X[/tex] the singleton set [tex]\{ x \}[/tex] is closed.

    Why is this the case? I can't see the reason from the definitions of the spaces.

    Let [tex](X,\tau)[/tex] be a topological space and let [tex]x,y \in X[/tex] be any two
    distinct points, if there exists any two open sets [tex]A,B \in \tau[/tex]
    such that [tex]x \in A[/tex] but [tex]x \notin B[/tex] and [tex]y \in B[/tex] but [tex]y \notin A[/tex],
    then [tex](X, \tau)[/tex] is a [tex]T_1[/tex] space.

    A topological space [tex](X, \tau)[/tex] is Hausdorff if for any
    [tex]x,y \in X[/tex], [tex]x \ne y[/tex], [tex]\exists \text{ neighborhoods } U \ni x[/tex] and
    [tex]V \ni y[/tex] such that [tex]U \cap V = \varnothing[/tex].
  2. jcsd
  3. Apr 17, 2010 #2


    User Avatar
    Science Advisor
    Homework Helper

    {x} is closed if the complement of {x} is open. Try to show X/{x} is open using the definitions. That's not so hard, is it?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook