Does a homeomorphism between two compact metric spaces preserve open balls? – math.stackexchange.com

Does a homeomorphism between two compact metric spaces preserve open balls? – math.stackexchange.com

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More specifically, if $X$ and $Y$ are compact metric spaces, and there is a $\phi: X \to Y$ a homeomorphism, then is it true that $\phi^{-1}(B(\phi(x),r))= B(x,r)$ ? If so, how? Thanks in advance!

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