I'm trying to prove that topological equivalence is an equivalence relation. Reflexivity was easy, and I'm sure transitivity is too, but I'm stuck on symmetry. My book's definition is that a metric space $(X,d_{1})$ is topologically equivalent to a metric space $(Y,d_{2})$ if there is a continuous bijection between them. But the inverse of a continuous bijection need not be continuous, so I'm not sure how to find a continuous bijection from $Y$ to $X$.
Continuity is given an epsilon-delta definition.
This is not true. Consider the continuous bijection from $[0,\pi)$ to $S^1$ given by $t \mapsto (\cos t,\sin t)$. Both are metric spaces, but there is no continuous bijection in the reverse direction since $S^1$ is compact but $[0,\pi)$ is not.
You could alternatively define $(X,d)$ and $(Y,d')$ to be equivalent if there are continuous bijections in both directions. This still doesn't make the spaces homeomorphic, but is certainly an equivalence relation.