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Re: Michael construction
Posted:
Mar 2, 2013 4:01 AM


On Thu, 28 Feb 2013, Butch Malahide wrote: > 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.
Indeed, a counter example is not surprizing. Is there a Hausdorff counter example?
A super Baire space (every intersection of denses sets is dense) is of the form S_D for some dense D and conversely for dense D, S_D is a super Baire space.
Is there any thing useful or outstanding about those spaces?



