  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.
