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X is compact set iff X closed and pre-compact. X is called pre-compact if $\overline{X}$ is compact. But in some texts, I found that X pre-compact is totally bounded. Is that two definitions of pre-compact have the same meaning?

Thanks for any help.

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1 Answer

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In a metric space $X$ compact implies $X$ closed. As then $X=\overline{X}$, $X$ is also precompact.

If $X$ is assumed to be precompact, so $\overline{X}$ is compact, and $X$ is closed too so that $X=\overline{X}$, $X$ is compact.

So the statement is trivial in any metric space.

$X$ precompact is equivalent to totally bounded in a complete metric space.

It's better to know the general metric fact $X$ is compact iff $X$ is complete and totally bounded.

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