A Distance-Preserving Bijection is a Homeomorphism

Let $(X,d_{X})$ and $(Y,d_{Y})$ be \reftext{def:metric-space-2026a}{metric spaces}. Equip $X$ with the collection $\mathcal{T}_{d_{X}}$ of all subsets that are \reftext{def:open-subset-metric-space-2026a}{open in $(X,d_{X})$}, which is a topology by \ref{thm:metric-open-sets-form-topology-2026a}, and equip $Y$ with the corresponding collection $\mathcal{T}_{d_{Y}}$. Write $B_{d}(x,r)$ for the \reftext{def:open-ball-metric-space-2026a}{open ball} with centre $x$ and radius $r$, and for a subset $A$ of $X$ write $\Phi(A)=\{\Phi(v):v\in A\}$.

Let $\Phi:X\to Y$ be a \reftext{def:bijection-sets-2026a}{bijection} that preserves distances, that is,

Then the following hold.

\textbf{1. (Balls correspond)} For every $x\in X$ and every real number $r>0$,

\textbf{2. (Homeomorphism)} Both $\Phi$ and its inverse $\Phi^{-1}:Y\to X$ are \reftext{def:continuous-map-topological-spaces-2026a}{continuous}.

\textbf{3. (Compactness)} For every $A\subseteq X$: the set $A$ is \reftext{def:compact-space-and-subset-2026a}{compact in $X$} if and only if $\Phi(A)$ is compact in $Y$.