---
date: '2024-11-27'
description: or topological isomorphism.
id: homeomorphism
modified: 2026-05-09 17:51:55 GMT-04:00
tags:
  - math
  - math/topology
title: homeomorphism
created: '2024-11-27'
published: '2024-11-27'
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slug: thoughts/homeomorphism
permalink: https://aarnphm.xyz/thoughts/homeomorphism.md
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---
alias: _topological isomorphism_, _bicontinuous function_

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

> \[!math\] 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)

> \[!tip\] `3^{\text{rd}}` requirements
>
> $f^{-1}$ is continuous is <mark>essential</mark>. 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)