Contractibility is not closure-preserved

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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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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.

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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 \R^n (via, for instance, a stereographic projection).