Vietoris thickenings and complexes of manifolds are homotopy equivalent

We show that if $X$ is a finite-dimensional Polish metric space, then the natural bijection $\mathrm{V}\mathrm{R}(X;r)\to \mathrm{V}{\mathrm{R}}^{\mathrm{m}}(X;r)$ from the (open) Vietoris-Rips complex to the Vietoris-Rips metric thickening is a homotopy equivalence. This occurs, for example, if $X$ is a Riemannian manifold. The same is true for the map $\check{\mathrm{C}}(X;r)$ to ${\check{\mathrm{C}}}^{\mathrm{m}}(X;r)$ from the Čech complex to the Čech metric thickening, and more generally, for the natural bijection $\mathrm{V}(\mathcal{W})\to {\mathrm{V}}^{\mathrm{m}}(\mathcal{W})$ from the Vietoris complex to the Vietoris metric thickening of any uniformly bounded cover $\mathcal{W}$ of a finite dimensional Polish metric space. We also show that if $X$ is a compact metrizable space, then ${\mathrm{V}}^{\mathrm{m}}(\mathcal{W})$ is strongly locally contractible.