  We study the relationship between metric thickenings and simplicial complexes
associated to coverings of metric spaces. Let $\mathcal{U}$ be a cover of a
separable metric space $X$ by open sets with a uniform diameter bound. The
Vietoris complex contains all simplices with vertex set contained in some $U
\in \mathcal{U}$, and the Vietoris metric thickening is the space of
probability measures with support in some $U \in \mathcal{U}$, equipped with an
optimal transport metric. We show that the Vietoris metric thickening and the
Vietoris complex have isomorphic homotopy groups in all dimensions. In
particular, by choosing the cover $\mathcal{U}$ appropriately, we get
isomorphisms between the homotopy groups of Vietoris--Rips metric thickenings
and simplicial complexes, where both spaces are defined using the convention
``diameter $< r$'' (instead of $\le r$). Similarly, we get isomorphisms between
the homotopy groups of \v{C}ech metric thickenings and simplicial complexes,
where both spaces are defined using open balls (instead of closed balls).
