Semi-locally simply connected

In mathematics, specifically algebraic topology, the semi-locally simply connected (or semilocally simply connected) property is a certain local connectedness condition that arises in the theory of covering spaces. Roughly speaking, a topological space X satisfies the property if for each point x in X any sufficiently small loop going through x can be contracted within X to a point. This condition is necessary for most of the theory of covering spaces, including the existence of a universal cover and the Galois correspondence between covering spaces and subgroups of the fundamental group.

Most “nice” spaces such as manifolds and CW complexes are semi-locally simply connected, and topological spaces that do not satisfy this condition are considered somewhat pathological. The standard example of a non-semi-locally simply connected space is the Hawaiian earring.