homeomorphism

alias: topological isomorphism, bicontinuous function

bijective and continuous function between topological spaces that has a continuous inverse functions.

definition

a function $f: X \rightarrow Y$ between two topological space is a homeomorphism if it has the following properties:

• $f$ is a bijection (one-to-one and onto)
• $f$ is continuous
• $f^{-1}$ as the inverse function is continuous (or $f$ is an open mapping)

$3^{\text{rd}}$ requirements

$f^{-1}$ is continuous is essential. Consider the following example:

• $f: \langle 0, 2 \pi ) \rightarrow S^1$ (the unit circle in $\mathbb{R}^2$) defined by $f(\varphi) = (\cos \varphi, \sin \varphi)$
  • is bijective and continuous
  • but not homeomorphism ($S^1$ is compact but $\langle 0, 2 \pi )$ is not)