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.