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Let $X$ be a Tychonoff countably compact space and $A$ is a subapce of $X$ such that for any countable $B \subset A$ we have $\overline{B} \subset A$. My question is this:

Is this subspace $A$ countably compact?

Thanks very muvh for any help.

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What if we let $A=X=\mathbb R$? –  Alex Becker Feb 25 '13 at 1:21

1 Answer 1

up vote 3 down vote accepted

Hint: Given an infinite $B \subseteq A \subseteq X$, every accumulation points of $B$ (as a subset of $X$) which belongs to $A$ is an accumulation point of $B$ viewed as a subset of $A$.

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