Topological quantum field theories are frequently obtained as quantum field theories which are constructed using some metric on the bordisms under consideration, but are then shown to be invariant under perturbations of the metric. This has led to the paradigm that topological quantum field theories are metric invariant quantum field theories. We turn this paradigm into a theorem in the \(\infty\)-categorical context by constructing a quasi-unital Segal space of Riemannian bordisms where the metric is allowed to vary continuously, and showing that the natural forgetful functor from Riemannian to ordinary bordisms is an equivalence. We will emphasise how on the one hand we need the Segal space technology to make the statement precise, but on the other hand the heart of the proof reduces to classical considerations in transversality theory.