Contractibility is not closure-preserved
This article gives the statement, and possibly proof, of a topological space property (i.e., contractible space) not satisfying a topological space metaproperty (i.e., closure-preserved property of topological spaces).
View all topological space metaproperty dissatisfactions | View all topological space metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for topological space properties
Get more facts about contractible space|Get more facts about closure-preserved property of topological spaces|
- Contractibility is not interior-preserved
- Seifert-van Kampen theorem
- Union of two simply connected open subsets with path-connected intersection is simply connected
Set to be a circle and set to be the complement of a single point in the circle. Then, is homeomorphic to the real line, and is contractible. However, it closure equals , which is not even simply connected, and therefore not contractible.
This example can be generalized to the -sphere for , where the complement of any point is homeomorphic to (via, for instance, a stereographic projection).