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Math and science::Topology

Homeomorphisms. Homeomorphic spaces.

Homeomorphisms and homeomorphic spaces.

Let \( X \) and \( Y \) be topological spaces.

Homeomorphism
A homeomorphism from \( X \) to \( Y \) is a [...].
Homeomorphic spaces
The spaces \( X \) and \( Y \) are homeomorphic iff [...].

\( X \cong Y \) is the notation for \( X \) to be homeomorphic to \( Y \). 'Topologically equivalent' is an alternative term for 'homeomorphic'.

When are two spaces homeomorphic?

To show that two spaces are homeomorphic, find a homeomorphism between them.

Roughly, two spaces are homeomorphic iff one can be deformed into the other by bending and reshaping, but without tearing or gluing.

Equivalence relation

Being homeomorphic is an equivalence relation on the class of all topological spaces.

Reflexive
Let \( X \) be a topological space. Then the identity map on \( X \) is a homeomorphism.
Symmetric
Let \( f : X \to Y \) be a homeomorphism. Then \( f^{-1} : Y \to X \) is a homeomorphism.
Transitive
Let \( f : X \to Y \) and \( g : Y \to Z \) be homeomorphisms. Then \( g \circ f : X \to Z \) is a homeomorphism.

Thus, \( X \cong X \); \( X \cong Y \iff Y \cong X \); \( X \cong Y \land Y \cong Z \implies X \cong Z \).