# 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*. 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.

## Definition

A space *X* is called **semi-locally simply connected** if every point in *X* has a neighborhood *U* with the property that every loop in *U* can be contracted to a single point within *X* (i.e. every loop is nullhomotopic). The neighborhood *U* need not be simply connected: though every loop in *U* must be contractible within *X*, the contraction is not required to take place inside of *U*. For this reason, a space can be semi-locally simply connected without being locally simply connected.

Equivalent to this definition, a space *X* is semi-locally simply connected if every point in *X* has a 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. In particular, this condition is necessary for a space to have a simply connected covering space.

## Examples

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. Give this space the subspace topology. Then all neighborhoods of the origin contain circles that are not nullhomotopic.

The Hawaiian earring can also be used to construct a semi-locally simply connected space that is not locally simply connected. In particular, the cone on the Hawaiian earring is contractible and therefore semi-locally simply connected, but it is clearly not locally simply connected.

Another example of a non-semi-locally simply connected space is the complement of **Q** × **Q** in the Euclidean plane **R**^{2}, where **Q** denotes the set of rational numbers. In fact, the fundamental group of this space is uncountable (Hatcher p. 54).

## Topology of fundamental group

In terms of the natural topology on the fundamental group, a locally path-connected space is semi-locally simply connected if and only if its quasitopological fundamental group is discrete.

## References

- Bourbaki, Nicolas (2016).
*Topologie algébrique: Chapitres 1 à 4*. Springer. Ch. IV pp. 339 -480. ISBN 978-3662493601. - J.S. Calcut, J.D. McCarthy
*Discreteness and homogeneity of the topological fundamental group*Topology Proceedings, Vol. 34,(2009), pp. 339–349 - Hatcher, Allen (2002).
*Algebraic Topology*. Cambridge University Press. ISBN 0-521-79540-0.