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If X×Y is Hausdorff, then is X Hausdorff? Why?

Jack

New member
Nov 8, 2012
9
If X Y is Hausdorff, then is X Hausdorff? Why?
If X Y is regular, then is X regular? Why?
 

Poirot

Banned
Feb 15, 2012
250
A set W is open in X x Y if for all (x,y) in W there exist open sets U in X, V in Y such that x is in u, y is in V, and U x V is a subset of W. So if X x Y is hausdorff, then for (x,y) and (x',y'), there exists disjoint open sets W and W' such that (x,y) is in W, (x',y') in W'. Thus, from above there exist open sets U and U' in X which must be disjoint ( otherwise a contradiction is obtained ) such that x is in u and x' is in U'.