Kelley 1.T, Brouwer reduction theorem

["Stephen J. Herschkorn"]

Problem 1.T in Kelley basically asks the reader to prove the following 
version of the Brouwer Reduction Theorem. Consider a topological space 
all of whose subspaces are Lindeloef.  Let  C  be a collection of closed 
sets with the property that if  D is a countable chain (with respect to 
inclusion) and a subfamily of  C,  then the intersection of  D  is in  
C.  Then for any  A  in  C,  there exists a minimal susbset  B  of  A  
which is also in  C.

First question:  My proof invokes Zorn's lemma.  Have I relied on the 
Axiom of Choice unnecessarily?

1.T(b)  asks,  in "an arbritrary topological space can any result of 
this general sort be affirmed?"  My answer is that one should remove the 
countable condition on  D  from the above hypothesis.  Is there a less 
restrictive hypothesis for this "sort" of result?

[BTW, this is not HW.]

--
Stephen J. Herschkorn                        sjherschko@xxxxxxxxxxxx
Math Tutor on the Internet and in Central New Jersey and Manhattan