Corollary

Let $(T(i))_{i \in I}$ be a family of subsets of $Y$ whose interiors cover $Y$, or which is a locally finite closed covering of $Y$. If $f : X \to Y$ is continuous and if each of the $f_{T(i)}$ is a homeomorphism of $f^{-1}(T(i))$ onto $T(i)$, then $f$ is a homeomorphism of $X$ onto $Y$.

For $f$ is clearly bijective, and is open by virtue of Proposition 2.