What Is a Homeomorphism in a Topological Space 📂Topology

What Is a Homeomorphism in a Topological Space

Definition ¹

For two topological spaces $X,Y$, if there exists a bijection $f : X \to Y$ such that both $f$ and its inverse function $f^{-1}$ are continuous functions, then $f$ is called a homeomorphism, and the two topological spaces are said to be homeomorphic.

Theorem

The following statements are equivalent.

(1): $f : X \to Y$ is a homeomorphism.
(2): $f^{-1} : Y \to X$ is a homeomorphism.
(3): $f : X \to Y$ is a continuous bijection that is a closed map.
(4): $f : X \to Y$ is a continuous bijection that is an open map.

Explanation

Just like what was defined in metric spaces, the concept of homeomorphism can also be extended in a simple way. It is fine to regard it as the very reason for studying continuous functions.

The reason (3) and (4), especially (4), are good is that they require no check on the inverse function. They are easily deduced from the properties of open maps and closed maps, so instead they satisfy the condition that the inverse function be continuous.

In particular, if both $f$ and $f^{-1}$ are differentiable, then $f$ is called a diffeomorphism.