A Distance-Preserving Bijection is a Homeomorphism

Let $(X,d_{X})$ and $(Y,d_{Y})$ be metric spaces. Equip $X$ with the collection $\mathcal{T}_{d_{X}}$ of all subsets that are open in $(X,d_{X})$, which is a topology by The Open Subsets of a Metric Space Form a Topology, and equip $Y$ with the corresponding collection $\mathcal{T}_{d_{Y}}$. Write $B_{d}(x,r)$ for the 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 bijection that preserves distances, that is,

$$d_{Y}(\Phi(u),\Phi(v))=d_{X}(u,v)\qquad\text{for all }u,v\in X .$$

Then the following hold.

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

$$\Phi\bigl(B_{d_{X}}(x,r)\bigr)=B_{d_{Y}}\bigl(\Phi(x),r\bigr).$$

2. (Homeomorphism) Both $\Phi$ and its inverse $\Phi^{-1}:Y\to X$ are continuous.

3. (Compactness) For every $A\subseteq X$: the set $A$ is compact in $X$ if and only if $\Phi(A)$ is compact in $Y$.