Re: Regular topological spaces


William Elliot wrote:
> From: Stephen J. Herschkorn <sjherschko@xxxxxxxxxxxx>
> Newsgroups: sci.math
> Subject: Regular topological spaces
>
> > Suppose a topological space has the property that any non-empty open
> > set contains the closure of a non-empty open set. Is the space
> > necessarily regular? Here, T1 is not part of the definition of
> > regularity.
>
> No. Open upper half plane with added x-axis where the open sets for the a
> point p on the x-axis is p with any open half disk centered at p. cf
> Steen's "Counterexamples in Topology" space 78 open half disk topology.

Alternatively: let X be an infinite set, choose a point p in X, and
call a subset of X open if it has a finite complement or it does not
contain p.

.

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