Semi-locally simply connected
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In mathematics, specifically algebraic topology, the phrase semi-locally simply connected refers to 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.
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[edit] 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). Note that 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.
[edit] Examples
Any “nice” space such as a manifold or CW complex is semi-locally simply connected. In some sense, only a pathological space can fail to satisfy this condition.
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 R2, where Q denotes the set of rational numbers.
[edit] Topology of fundamental group
In terms of the natural topology on the fundamental group, a space is semi-locally simply connected if and only if its topological fundamental group is discrete.
[edit] References
- Daniel K. Biss, A Generalized Approach to the Fundamental Group The American Mathematical Monthly, Vol. 107, No. 8 (Oct., 2000), pp. 711–720
- Hatcher, Allen (2002). Algebraic Topology. Cambridge University Press. ISBN 0-521-79540-0. http://www.math.cornell.edu/~hatcher/AT/ATpage.html.
