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From: Stephen J. Herschkorn <sjherschko@xxxxxxxxxxxx>
Newsgroups: sci.math
Subject: Kelley 1.T, Brouwer reduction theorem
> 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.
Kelley states this poorly as he makes no connection between the property P
and P being an inductive property in the theorem, other than by spatial
association and psychic surmise.
> First question: My proof invokes Zorn's lemma. Have I relied
> on the Axiom of Choice unnecessarily?
Don't think so.
> 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?
Besides countable, also removable from the notion of inductive, is nest
and/or closed for property P. With nest removed, AxC isn't needed. With
countable removed, inductive is somewhat a misnomer and the strength of
the theorem diminished.
The answer to 1.T(b) is no. An example of a space that Brower's reduction
theorem doesn't apply is omega_1.
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