Definition:Homeomorphism

Topological Spaces
Let $$T$$ and $$T'$$ be topological spaces.

Let $$f: T \to T'$$ be a bijection such that both $$f$$ and $$f^{-1}$$ are continuous.

Then $$f$$ is a homeomorphism. We can say that $$T$$ and $$T'$$ are homeomorphic.

Equivalently, by definition of continuity, a homeomorphism is a bijection such that $$U$$ is open in $$T$$ iff $$f \left({U}\right)$$ is open in $$T'$$.

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

Let $$f: M \to M'$$ be a bijection such that both $$f$$ and $$f^{-1}$$ are continuous.

Then $$f$$ is a homeomorphism.

This definition also follows directly from:
 * The fact that a metric space induces a topology;
 * Equivalence of Metric Space Continuity Definitions.

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.

Caution
Not to be confused with homomorphism.