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.