Re: Hausdorff and compact

On Sat, 22 Apr 2006 18:48:44 -0700, William Elliot
<marsh@xxxxxxxxxxxxxxxxxx> wrote:


(...) In general I doubt that
every topology contains a minimal Hausdorff topology. Namely as
you see, there's a problem applying Zorn's lemma. (...)

What you doubt is indeed false. I find in the book suggested by
G. A. Edgar (Handbook of the history of general topology, Vol. 2)
in the chapter on minimal Hausdorff spaces the fact that the usual
topology on the set of rationals isn't finer than any minimal
Hausdorff topology (content of result 2.2 on p.678 without
using the extra expressions defined there) - this is stronger
than the fact mentioned and proven in this thread that |Q is
an example of what the OP asked (reduced to considering
identity map, i.e. its top. isn't finer than any compact H. topology)

This implies that there exists a totally ordered set S of Hausdorff
topologies on |Q coarser than the usual one, such that the
intersection of S isn't Hausdorff. May-be even a decreasing sequence ?

BTW the same book chapter also contains an example of
a countable non-compact minimal Hausdorff space. Therefore
the set |Q itself has a non-compact minimal Hausdorff topology
which of course is not comparable (for inclusion partial order)
with the usual one. I haven't seen yet the book "Counterexamples
in Topology", so I can't say whether this example is essentially
the same as #100 in that, but it somehow replaces it for me ;-)
.

Relevant Pages

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  • Re: compact Hausdorff topology on R
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  • Re: compact Hausdorff topology on R
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  • Re: Hausdorff and compact
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  • Re: Hausdorff and compact
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