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 ''x'' in ''X'' and every neighborhood ''V'' of ''x'' has an open neighborhood ''U'' of ''x'' such that $x \in U \subset V$ with the property that every loop in ''U'' can be contracted to a single point within ''X'' (i.e. every loop in ''U'' is nullhomotopic in ''X''). 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 called semi-locally simply connected if every point in ''X'' has a open neighborhood ''U'' with the property that every loop in ''U'' can be contracted to a single point within ''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). 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.

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

*
* J.S. Calcut, J.D. McCarthy ''Discreteness and homogeneity of the topological fundamental group'' Topology Proceedings, Vol. 34,(2009), pp. 339–349
*{{cite book , first = Allen , last = Hatcher , author-link = Allen Hatcher , year = 2002 , title = Algebraic Topology , publisher = Cambridge University Press , isbn = 0-521-79540-0 , url = http://pi.math.cornell.edu/~hatcher/AT/ATpage.html

Algebraic topology
Homotopy theory
Properties of topological spaces