Closed continuous surjective map and Hausdorff space

  1. radou

    radou 3,108
    Homework Helper

    1. The problem statement, all variables and given/known data

    Here's a nice one. I hope it's correct.

    Let p : X --> Y be a closed, continuous and surjective map such that p^-1({y}) is compact for every y in Y. If X is Hausdorff, so is Y.

    3. The attempt at a solution

    Let y1 and y2 in Y. p^-1({y1}) are then p^-1({y2}) disjoint and compact subsets of X. Since X is Hausdorff, for p^-1({y1}) and for any x in p^-1({y2}) there exist disjoint open sets U and V containing p^-1({y1}) and x, respectively. Now find such pair of open sets for p^-1({y1}) and for any x in p^-1({y2}). These sets form open covers for p^-1({y1}) and for p^-1({y2}) respectively, so they have finite subcovers. Take the intersection of all sets from the finite subcover for p^-1({y1}), let's call it U1. Take the union of all sets from the finite subcover for p^-1({y2}), call it U2. U1 and U2 are disjoint.

    Now, since U1 and U2 are open sets containing p^-1({y1}) and p^-1({y2}) respectively, there exist neighborhoods W1 of y1 and W2 of y2 such that p^-1(W1) is contained in U1 and p^-1(W2) is contained in U2.

    I claim that W1 and W2 are disjoint.

    Suppose they were not - let y be an element in their intersection. Then p^-1({y}) is contained both in U1 and U2, contradicting the fact that U1 and U2 are disjoint.
  2. jcsd
  3. micromass

    micromass 20,039
    Staff Emeritus
    Science Advisor
    Education Advisor

    Your proof is perfect!
  4. radou

    radou 3,108
    Homework Helper

    Excellent! This tradition mustn't go on, since I'll start to think I'm good :D

    Btw, the "hint" is extremely useful. I don't see another way we could "generate" an open set with certain required properties in the codomain.
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