Math and science::Topology
Homeomorphisms. Homeomorphic spaces.
Homeomorphisms and homeomorphic spaces.
Let
- Homeomorphism
- A homeomorphism from
to is a [...]. - Homeomorphic spaces
- The spaces
and are homeomorphic iff [...].
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
be a topological space. Then the identity map on is a homeomorphism. - Symmetric
- Let
be a homeomorphism. Then is a homeomorphism. - Transitive
- Let
and be homeomorphisms. Then is a homeomorphism.
Thus,