Homeomorphism

Not to be confused with Homomorphism

Homeomorphism of metric spaces

Given two metric spaces $(X,d)$ and $(Y,d')$ they are said to be homeomorphic if:

• There exists a mapping $f:(X,d)\rightarrow(Y,d')$ such that:
  1. $f$ is bijective
  2. $f$ is continuous
  3. $f^{-1}$ is also a continuous map

Then $(X,d)$ and $(Y,d')$ are homeomorphic and we may write $(X,d)\cong(Y,d')$ or simply (as Mathematicians are lazy) $X\cong Y$ if the metrics are obvious

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TODO: Find reference for use of $\cong$ notation

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Topological Homeomorphism

A topological homeomorphism is bijective map between two topological spaces

$$f:(X,\mathcal{J})\rightarrow(Y,\mathcal{K})$$ where:

1. $$f$$ is bijective
2. $$f$$ is continuous
3. $$f^{-1}$$ is continuous

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TODO: Using Continuity definitions are equivalent it is easily seen that the metric space definition implies the second, that logic and a reference would be good!

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See also

• Composition of continuous maps is continuous