Homotopy in Topological Spaces 📂Topology

Homotopy in Topological Spaces

Definition ¹

For two topological spaces $X,Y$, if a bijection $f : X \to Y$ exists 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 propositions 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 function.
(4): $f : X \to Y$ is a continuous bijection that is an open function.

Explanation

Just as with the concept defined in the metric space, the concept of homeomorphism can also be easily extended. It is also reasonable to see it as the very reason to study continuous functions.

The particularly good reason for (3) and especially (4) is because it eliminates the need for checking the inverse function. It’s easily deduced from the properties of open and closed functions, satisfying the condition that the inverse function must be continuous in their stead.

Especially, if $f,f^{-1}$ is differentiable, it is called a Diffeomorphism.