Semi-locally simply connected

From Academic Kids

In mathematics, in particular topology, a topological space X is called semi-locally simply connected if every point x in X has a neighborhood U such that 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. That is, every loop in U can be deformed to a point. This is true of the 'best' spaces such as manifolds and simplicial complexes.

Evidently, a space that is locally simply connected is semi-locally simply connected. An 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. Give this space the subspace topology. Then all neighborhoods of the origin contain circles that are not nullhomotopic.

The property of semi-locally simple connectivity is weaker than that of local simple connectivity. To see this, consider the cone on the Hawaiian earring. It is contractible and therefore semi-locally simply connected, but it is clearly not locally simply connected.

In the theory of covering spaces, a space has a universal cover if and only if it is path-connected, locally path-connected, and semi-locally simply connected.

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