Let $X$ be a (irreducible) variety, $\tilde{X}$ its normalization, $\pi\colon \tilde{X}\to X$ the natural map. Is there a "nice" characterization of when $\pi$ is a homeomorphism? E.g. is it enough to know that $\pi$ is injective?

If it makes things simpler, I'm most interested in the case that $X$ is an affine toric variety over $\Bbb C$.