1. Not finding help here? Sign up for a free 30min 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!

Topology, projection map

  1. Nov 11, 2004 #1
    I'm trying to prove some stuff that involves the projection map, say p:X x Y ->X. But I need to know if it's continuous. If a map is continuous, then the preimage of a open/closed set is open/closed.

    The problem is, what do open sets in X x Y look like? I know what the basis elements are, and the open sets would be arbitrary unions and finite intersections, but is there any way to generalize?
     
  2. jcsd
  3. Nov 11, 2004 #2

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    You don't need to know all open sets, you just need to show that some particular sets are open.
     
  4. Nov 11, 2004 #3
    Well, yeah, but I thought that if there was a way to list all the open sets, that would take care of it.

    Ok, if I take p(A) to be open, I need to show that the preimage is open. The preimage would be A x B, where A is open in X. If B is open, then I'm done. But if B is closed, I don't know what to do.
     
  5. Nov 12, 2004 #4
    So [tex]p: X \times Y \to X[/tex] and [tex]p((x,y)) = x[/tex]. But now think about it what sets will give you [tex]\{x\}[/tex] as an image? [tex]p((x,y)) = x[/tex] for all [tex]y \in Y[/tex]. So [tex]p^{-1}(\{x\}) = \{x\} \times Y[/tex]. I think you can take it from there.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Topology, projection map
  1. Topology question (Replies: 5)

  2. Continuous Mappings (Replies: 30)

  3. Mappings problem (Replies: 1)

Loading...