
Re: Michael construction
Posted:
Mar 1, 2013 2:41 AM


On Feb 28, 10:27 pm, William Elliot <ma...@panix.com> wrote: > Let M be a subset of a normal space S. Define S_M as S given the > base { U, {x}  U open within S, x in M } (Michael construction). > > If in addition, S\M is normal, is S_M normal?
Let S = omega. Let the open subsets of S be the empty set, the whole space S, and those cofinite subsets of S which do not contain 0. Let M = S \ {1, 2}. S is normal, because every nonempty closed set contains 0. S \ M is normal, because it's discrete in its relative topology. S_M is not normal, because it's a T_1space but is not Hausdorff.

