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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. $\endgroup$ – user38355 Aug 6 '13 at 6:35
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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.

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