Closed continuous surjective map and Hausdorff spaceby radou Tags: continuous, hausdorff, space, surjective 

#1
Nov2210, 02:13 PM

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P: 3,225

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. 


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