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Definition: Homeomorphism, Homeomorphic Spaces

Let \(X\) and \(Y\) be topological spaces. A function \(\varphi \colon X\longrightarrow Y\) is called an homeomorphism, if

• \(\varphi\) is continuous and
• \(\varphi\) is bijective and
• the inverse \(\varphi ^{-1}\) is also continuous.

Two sets $X$ and $Y$ are homeomorphic (or topologically equivalent), if there exists a homeomorphism \(\varphi \colon X\longrightarrow Y\) between them.

Notes

• Homeomorphic sets are indistinguishable from a topological point of view since every open set of $X$ corresponds to exactly one open set of $Y$ and vice versa.
• However, it is possible to have homeomorphic spaces $X$ and $Y$ with different topologies.
• It is also possible that $X$ and $Y$ are homeomorphic, and have subsets $A\subset X$ and $B\subset Y$ that are also homeomorphic with each other, but there is no homomorphism of $X$ and $Y$ taking $A$ onto $B.$ For instance, take the subsets of real numbers $A=\{0\}\cup[1,2]\cup\{3\}$ and $B=[0,1]\cup \{2\}\cup \{3\}.$
• Moreover, it is not automatically the case that if $f$ is continuous and bijective, then $f^{-1}$ is continuous. For instance, if $X$ has discrete topology $\mathcal O_X$ and $Y$ has an indiscrete topology $\mathcal O_Y,$ then the identity function $id:X\to Y$ is continuous and bijective, but the inverse $id^{-1}$ is not continuous.

Mentioned in:

Definitions: 1 2 3 4
Proofs: 5
Propositions: 6

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References

Adapted from CC BY-SA 3.0 Sources:

1. Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of Osnabrück