From: Fred Galvin <galvin@math.ukans.edu>
Subject: Re: Hard Topology Problem
Date: Fri, 14 Jan 2000 01:30:15 -0600
Newsgroups: sci.math
Summary: [missing]

On Fri, 14 Jan 2000, Jeff Streets wrote:

> Can anyone assert the truth or falsehood of the following, 
> or give a condition on a topological space X for which the
> statement is true?
> 
> If (X, T) a topological space and S a dense subset of X which
> is normal in the subspace topology, is X also normal?

I assert the falsehood. Every example I can think of, of a non-normal
regular space, has a normal dense subspace. For example, Niemytzki's
Tangent Disc Topology on the closed upper half-plane is a non-normal
Tychonoff space, and the open upper half-plane (with the usual metric
topology) is a dense subspace. For another example, let X be the
Sorgenfrey plane, and let S be any countable dense subset of X.

Do you know an example of a space X which does *not* have a normal
dense subspace?
==============================================================================

From: Fred Galvin <galvin@math.ukans.edu>
Subject: Re: Hard Topology Problem
Date: Fri, 14 Jan 2000 17:58:45 -0600
Newsgroups: sci.math

On Fri, 14 Jan 2000, K. P. Hart wrote:

> Fred Galvin wrote:
> 
> > Do you know an example of a space X which does *not* have a normal
> > dense subspace?
> 
> Now *that* is a stumper!

A colleague of mine has an example of a completely Hausdorff space in
which no dense subspace is semiregular: Jack Porter, "A Hausdorff
space with no dense regular subspace", preprint, 1995.