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

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William Elliot

Posts: 1,707
Registered: 1/8/12
Re: Michael construction
Posted: Mar 2, 2013 4:01 AM
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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_1-space 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?



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