Re: General topology question

From: Stuart M Newberger <smnewberger@xxxxxxxxxxx>
Newsgroups: sci.math
Subject: General topology question

> Hocking and Young strike again

> They define countable compactness by -every infinite subset has a
> limit point which means every neighborhood has at least one other
> point from the subset.

almost countably compact

> This is weaker than w-limit point which means every nbd has
> infinitely many points of the set(they are equivalent if the
> space is T_1).

> Requiring w-limit point for infinite subsets is equivalent to saying
> that every countable open cover has a finite subcover and this is
> this is equivalent to the definition of countable compactness in eg
> J Kelly-General Topology and probably what most people use these
> days.

countably compact ==> almost countably compact.
T0 counterexample for converse:
Let N have topology { [1,n] /\ N | n in N } \/ { nulset }
N not countably compact. If A infinite subset N, let n = min A.
For all j in A\n, j limit point. Thus N almost countably compact.

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