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Homework Help: 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


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    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.
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