Vietoris thickenings and complexes are weakly homotopy equivalent

Characterizing the homotopy types of the Vietoris--Rips complexes of a metric space $X$ is in general a difficult problem. The Vietoris--Rips metric thickening, a metric space analogue of the Vietoris--Rips complex, was introduced as a potentially more amenable object of study with several advantageous properties, yet the relationship between its homotopy type and that of the Vietoris--Rips complex was not fully understood. We show that for any metric space $X$ and threshold $r>0$, the natural bijection between the (open) Vietoris--Rips complex and Vietoris--Rips metric thickening is a weak homotopy equivalence.