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Topic: Michael construction
Replies: 2   Last Post: Mar 2, 2013 4:01 AM

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Butch Malahide

Posts: 894
Registered: 6/29/05
Re: Michael construction
Posted: Mar 1, 2013 2:41 AM
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On Feb 28, 10:27 pm, William Elliot <> 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_1-space but is not Hausdorff.

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