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

Vietoris thickenings and complexes are weakly homotopy equivalent