Homeomorphism

A homeomorphism is an isomorphism in $𝖳 𝗈 𝗉$ , i.e. a continuous map with a continuous inverse. #m/def/topology Therefore both images of open sets in the domain are open in the codomain, and preïmages of open sets in the codomain are open in the domain. Homeomorphisms preserve every Topological property.

Unlike with categories of algebras, a bijective continuous map is not necessarily a homeomorphism, there exist continuous bijections with non-continuous inverses. However, A continuous bijection from compact to Hausdorff is a homeomorphism.

Example

Consider two topologies $T 1$ and $T 2$ on $𝑋$ , where $T 1 ⊊ T 2$ (see Coarseness and fineness of topologies). Then the identity map $𝑖 : (𝑋, T 1) \to (𝑋, T 2) : 𝑥 \mapsto 𝑥$ is continuous and bijective, but its inverse $𝑖 - 1$ is not. Therefore, $𝑖$ is not a homeomorphism.

Properties

A map is a homeomorphism iff it is bijective, continuous, and open

Homeomorphism

Properties

See also