From: Dave Rusin <rusin@math.niu.edu>
Subject: Re:  Some things to ponder
Date: Thu, 17 Aug 2000 14:42:35 -0500 (CDT)
To: richard@math.niu.edu
Summary: [missing]

You asked,
>if a topological space X has the property that every nested (infinite) 
>sequence of closed sets has a nonempty intersection, does it follow
>that X is compact?

Theorem: For a topological space  X  the following are equivalent:
(1) for every sequence  F1 \contains  F2 \contains ...  of nonempty 
    closed subsets  F_i  we have  \intersect F_i  \not= \emptyset
(2) X  is countably compact

Proof: (1) => (2) : 

Suppose given a countable open cover { U_i, i=1, 2, ... } . Define
F1 = X - U_1
F2 = X - U_1 - U_2
F3 = X - U_1 - U_2 - U_3
etc.  Then the  F_i  are closed, with  F_i  \subseteq  F_{i-1}. 
We have  \intersect F_i = X - \union U_i = \emptyset, so from (1) we
deduce that the  F_i  are NOT all nonempty. But if some  F_n = \emptyset
then   {U_1, ..., U_n}  is a cover of  X.  

So every countable open cover has a finite subcover, so X is countably compact.


(2) => (1) : 

Given a nested sequence of nonempty closed subsets  F_i  we let
U_i = X - F_i, a proper open subset of  X. Now, if  \intersect F_i  were
empty, then the  U_i  would form a countable cover of  X, and then (2)
would mean there would be a finite subcover, although the nesting
of the  U_i  would then force a single U_n  to equal  X, a contradiction.
So the intersection of the  F_i  cannot be empty.

QED


Corollary: (1)  is NOT equivalent to compactness.
Proof: there are countably compact, non-compact spaces. (They cannot be
Lindeloef or paracompact, though.) An example is the topological space
derived from the ordered set  [0, \omega_1)  (where  \omega_1  is the
first uncountable ordinal). It's not compact (consider the open sets
[0, alpha) where  alpha  ranges over the countable ordinals) but it is 
countably compact as can be seen by using the theorem above: from a 
_countable_ collection of closed sets one can find something in the 
intersection by mimicking Cantor's nested interval theorem, using the fact that
_countable_ sets of increasing ordinals have a supremum less than \omega_1.