How to prove,in topological space,every compact space is separable.?

Question

Please write in detail.

Answer 1

I'm afraid Charles is right. This is true of metric spaces, however. However, using Lindelof's theorem, any topological space that has a COUNTABLE basis for the open sets, is then separable, but is only compact if it satisfies the Bolzano-Weierstrass property. In other words, if a set has a countable basis for the open sets (or, as some put it, satisfies the second axiom of countability), then your statement is trivially correct. Most counter-examples here are of the type that have uncountable bases AND are non-metrizable. Hope this helps.

It is perhaps easy to visualize a counterexample, since compactness is preserved under uncountable products, but separability is not. Simply take the aleph-2 product of Z+, which is clearly compact and therefore separable. The product is still compact, but no longer is it separable.


Steve

Answer 2

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Answer 3

This is *not* so. Here is a small list of counter examples:

http://at.yorku.ca/cgi-bin/bbqa?forum=ask_a_topologist_2003;task=show_msg;msg=0014.0001

HTH

Charles