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 equip...