# Contractibility is not closure-preserved

From Topospaces

This article gives the statement, and possibly proof, of a topological space property (i.e., contractible space)notsatisfying a topological space metaproperty (i.e., closure-preserved property of topological spaces).

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## Definition

It is possible to have a topological space and a subset of such that is a contractible space in the subspace topology from , but , the closure of in , is not contractible.

## Related facts

- Contractibility is not interior-preserved
- Seifert-van Kampen theorem
- Union of two simply connected open subsets with path-connected intersection is simply connected

## Proof

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).