Re: number of topologies on all non-empty subsets of R

David Bernier wrote (in part):

The equivalence classes consist of subsets of R which are
homeomorphic in the metric topology from R. So P(R) has
2^(2^aleph_0) elements. I'm wondering how many equivalence
classes of topologies we have.

I believe the answer is 2^(2^aleph_0), even for a weaker relation
known as "dimensional type". See the references I made to Sierpinski's
topology book in this post, especially p. 131 of Sierpinski's book:

http://groups.google.com/group/sci.math/msg/25de172ecd508813

Dave L. Renfro

.

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