Absolute neighborhood retract

Summary

An absolute neighborhood retract, or ANR, is a metrizable space with the following property. Whenever it sits as a closed subset in a metrizable space, it is a retract of some neighbourhood of itself. ANRs include manifolds, polyhedra, open subsets of Banach spaces and convex subsets of normed spaces. They are the spaces with good local behaviour for homotopy theory. They are locally contractible, and compact ANRs have the homotopy type of finite CW complexes.