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