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Discrete metric space

Posted: Wed Nov 25, 2015 7:27 pm
by Papapetros Vaggelis
Prove that a metric space \(\displaystyle{\left(X,d\right)}\) containing infinite points, where \(\displaystyle{d}\)

is the discrete metric, is not compact.

Re: Discrete metric space

Posted: Thu Nov 26, 2015 12:38 am
by Nikos Athanasiou
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.