Semi-locally simply connected

In mathematics, specifically algebraic topology, semi-locally simply connected is a certain local connectedness condition that arises in the theory of covering spaces.

Roughly speaking, a topological space X is semi-locally simply connected if there is a lower bound on the sizes of the “holes” in X.

Another equivalent way to define this concept is the following, a space X is semi-locally simply connected if every point in X has an open neighborhood U for which the homomorphism from the fundamental group of U to the fundamental group of X, induced by the inclusion map of U into X, is trivial.

Most of the main theorems about covering spaces, including the existence of a universal cover and the Galois correspondence, require a space to be path-connected, locally path-connected, and semi-locally simply connected, a condition known as unloopable (délaçable in French).

A simple example of a space that is not semi-locally simply connected is the Hawaiian earring: the union of the circles in the Euclidean plane with centers (1/n, 0) and radii 1/n, for n a natural number.