# Difference between revisions of "Homeomorphism"

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− | Let $(X,\tau)$ and $(Y,\sigma)$ be [[topological spaces]]. Let $f \colon X \rightarrow Y$ be a [[function]]. We say that $f$ is a ''homeomorphism between $(X,\tau)$ and $(Y,\sigma)$'' if the following properties are satisfied: | + | Let $(X,\tau)$ and $(Y,\sigma)$ be [[topological space|topological spaces]]. Let $f \colon X \rightarrow Y$ be a [[function]]. We say that $f$ is a ''homeomorphism between $(X,\tau)$ and $(Y,\sigma)$'' if the following properties are satisfied: |

# $f$ is a [[function | bijection]]; | # $f$ is a [[function | bijection]]; | ||

# $f$ is [[continuous]]; | # $f$ is [[continuous]]; | ||

# the inverse function function $f^{-1}$ is continuous. | # the inverse function function $f^{-1}$ is continuous. |

## Latest revision as of 01:10, 30 April 2015

Let $(X,\tau)$ and $(Y,\sigma)$ be topological spaces. Let $f \colon X \rightarrow Y$ be a function. We say that $f$ is a *homeomorphism between $(X,\tau)$ and $(Y,\sigma)$* if the following properties are satisfied:

- $f$ is a bijection;
- $f$ is continuous;
- the inverse function function $f^{-1}$ is continuous.