Vietoris thickenings and complexes have isomorphic homotopy groups

We study the relationship between metric thickenings and simplicial complexes associated to coverings of spaces. Let $${\mathcal {U}}$$ be a cover separable space X by open sets with uniform diameter bound. The Vietoris complex {V}}({\mathcal {U}})$$ contains all simplices vertex set contained in some $$U \in {\mathcal , thickening {V}}^\textrm{m}({\mathcal is probability measures support equipped an optimal transport metric. show that have isomorphic homotopy groups dimensions. In particular, choosing appropriately, we get isomorphisms Vietoris–Rips $$\pi _k(\textrm{VR}^\textrm{m}(X;r))\cong \pi _k(\textrm{VR}(X;r))$$ for integers $$k\ge 0$$ where both spaces are defined using convention “diameter $$< r$$ ” (instead $$\le ). Similarly, Čech _k(\check{\mathrm {{C}}}^\textrm{m}(X;r))\cong {{C}}}(X;r))$$ balls closed balls).