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!

Product topology, closed subset, Hausdorff

  1. Apr 26, 2010 #1
    1. The problem statement, all variables and given/known data

    Let [tex](X,\tau_X)[/tex] and [tex](Y,\tau_Y)[/tex] be topological spaces, and let [tex]f : X \to Y[/tex] be continuous. Let [tex]Y[/tex] be Hausdorff, and prove that the graph of [tex]f[/tex] i.e. [tex]\graph(f) := \{ (x,f(x)) | x \in X \}[/tex] is a closed subset of [tex]X \times Y[/tex].

    2. Relevant equations

    3. The attempt at a solution

    Which property of closed set should I use to prove this? Should I assume a sequence inside the graph set converging to some [tex](x,y) \in X[/tex] and then somehow show that this limit point belongs to the graph? Or should I prove that the complement of the graph set is not open? I don't know how to finish the proof with either approach. Please give me some hint.
  2. jcsd
  3. Apr 26, 2010 #2
    The idea here is to find (x,y) not on the graph such that every neighbourhood misses the graph. To do this, pick (x,y) not on the graph, so it is different than (x,f(x)) and seperate them by neighbourhoods. You must also use continuity here.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Product topology, closed subset, Hausdorff
  1. Closed subsets (Replies: 5)