From: Robin Chapman <rjc@maths.ex.ac.uk>
Subject: Re: countable compactness / finite intersection
Date: Wed, 29 Mar 2000 09:23:07 GMT
Newsgroups: sci.math

In article <38E1168D.3E475C39@cbrownsystems.com>,
  Chas F Brown <cbrown@cbrownsystems.com> wrote:
>
>
> Robin Chapman wrote:
> >
> > In article <38DEF14D.6136B8B9@cbrownsystems.com>,
> >   Chas F Brown <cbrown@cbrownsystems.com> wrote:
> > >
>
> > > "S is countably compact" is equivalent to "if C = {C_n} is a countable
> > > collection of closed sets in S satisfying the finite intersection
> > > hypothesis, then (intersection {C_n}) is non-empty".
> > >
> > > My approach was essentially to make an infinite set containing points of
> > > the successive intersections of sets in C, which works fine if the
> > > singleton set {p} never has a limit point. But I feel like I have a
> > > counter-example; can you help me with my "cognitive bug"?
> > >
>

 <snip of proof that...>
>
> > > This contradicts the equivalence I was supposed to prove.
>
> > Looks fine to me. You should instead find the weakest separation axiom
> > that saves the result!
>
> I needed a little confidence that I understood correctly how to
> construct these topologies first! It seems that T1 spaces fit the bill,
> but certainly a basis on Z (*all* integers!) of "half-planes" B_i = {x |
> x<=i} has every (p a limit point of q) iff (p > q), and so is T0 and

Do you mean p < q here?

> countably compact, but again, the collection of closed sets (Z - B_i)
> for i>0 satisfies the finite intersection hypothesis but has overall
> empty intersection.
>
> I think (i.e, haven't proven that...) the overall requirement is that
> every closed set C must contain a point p s.t. {p} has no limit point.
> Of course, some T0 spaces DO meet this requirement, but not all.

But every T1 space does. Are there such things as T 1/2 spaces?
Where's my copy of Steen and Seebach? .... :-)

Anyway, you should probably use the term "countably compact" for the
second property, that each countable family of closed with finite
intersection property has nonempty intersection (equivalently,
each countable open cover has a finite subcover).

> Thanks as ever for your feedback, Robin. Don't you get tired of looking
> at elementary stuff?

Something slightly less elementary might be Hocking and Young's "proof"
of the nontrivialilty of the Hopf fibration (p. 186). I'm not convinced
by this. For a start I'm sure their equations for x_1, x_3 and x_4
in terms of alpha and beta are wrong, and in any case this would
not extend to a continuous function y on S^2 (which they need).
The function y would fail to be continuous at one of the "poles"
of the sphere. Can *this* proof be rescued?

--
Robin Chapman, http://www.maths.ex.ac.uk/~rjc/rjc.html
 "`The twenty-first century didn't begin until a minute
  past midnight January first 2001.'"
   John Brunner, _Stand on Zanzibar_ (1968)


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