we know that each infinite subspace of a KC space contain an infinite discrete subspace
The Hausdorff spaces imply KC spaces.But:
Is it right to say that every Hausdorff space has an infinite discrete subspace?
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$\begingroup$ Any finite discrete space is Hausdorff, but obviously can't contain an infinite discrete subspace. $\endgroup$– user38355Aug 6, 2013 at 6:35
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1 Answer
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It is true that every infinite Hausdorff space $X$ has an infinite discrete (in itself) subspace. The extra condition that $X$ is infinite itself is clearly necessary... And if you know the fact for KC spaces, it certainly follows for Hausdorff spaces.
answered Aug 6, 2013 at 7:23
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$\begingroup$ I want to see exact proof for Hausdorff space, please tell me which book show it? Thanks. $\endgroup$– habibAug 6, 2013 at 7:45
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