Contractibility is not closure-preserved

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