Definition:Homeomorphism

Metric Spaces
Let $$M$$ and $$M'$$ be metric spaces.

Let $$f: M \to M'$$ be a bijection such that both $$f$$ and $$f'$$ are continuous.

Then $$f$$ is a homeomorphism.

Manifolds
A homeomorphism of a manifold $$X$$ to a manifold $$Y$$ is a continuous bijection such that the inverse is also continuous.

Note
Also known as a topological equivalence.