Editing Homeomorphism (section)

==Properties==
* Two homeomorphic spaces share the same [[topological property|topological properties]]. For example, if one of them is [[compact space|compact]], then the other is as well; if one of them is [[connectedness|connected]], then the other is as well; if one of them is [[Hausdorff space|Hausdorff]], then the other is as well; their [[homotopy]] and [[homology group]]s will coincide. Note however that this does not extend to properties defined via a [[metric space|metric]]; there are metric spaces that are homeomorphic even though one of them is [[completeness (topology)|complete]] and the other is not.
* A homeomorphism is simultaneously an [[open mapping]] and a [[closed mapping]]; that is, it maps [[open set]]s to open sets and [[closed set]]s to closed sets.
* Every self-homeomorphism in <math>S^1</math> can be extended to a self-homeomorphism of the whole disk <math>D^2</math> ([[Alexander's trick]]).