Definition:Homeomorphism/Topological Spaces

Definition
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.

$T$ and $T'$ are said to be homeomorphic.

The symbol $T \sim T'$ is often seen.

Also see

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


 * By Bijection is Open iff Inverse is Continuous a homeomorphism is a bijection which is both open and continuous.


 * By Bijection is Open iff Closed it follows that a homeomorphism is a bijection which is both closed and continuous.

Also known as
Also known as:
 * a topological equivalence, usually used when the spaces in question are metric spaces
 * an isomorphism.

Caution
Not to be confused with homomorphism.

Also see

 * Inverse of Homeomorphism is Homeomorphism