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Discrete metric space

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Papapetros Vaggelis
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Discrete metric space

Prove that a metric space $\displaystyle{\left(X,d\right)}$ containing infinite points, where $\displaystyle{d}$

is the discrete metric, is not compact.
Nikos Athanasiou
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Re: Discrete metric space

In a discrete metric space, all points are open sets. Take the open cover $$\lbrace \lbrace x \rbrace \mid x \in X \rbrace$$.

This does not have a finite subcover, since $X$ contains infinitely many points.