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).
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Definition

It is possible to have a topological space $X$ and a subset $A$ of $X$ such that $A$ is a contractible space in the subspace topology from $X$, but $\overline{A}$, the closure of $A$ in $X$, 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 $X$ to be a circle and set $A$ to be the complement of a single point in the circle. Then, $A$ is homeomorphic to the real line, and is contractible. However, it closure $\overline{A}$ equals $X$, which is not even simply connected, and therefore not contractible.

This example can be generalized to the $n$-sphere $S^n$ for $n\ge 1$, where the complement of any point is homeomorphic to ${\mathbb{R}}^n$ (via, for instance, a stereographic projection).