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Semilocally Simply Connected


A topological space X is semilocally simply connected (also called semilocally 1-connected) if every point x in X has a neighborhood U such that any loop L:[0,1]->U with basepoint x is homotopic to the trivial loop. The prefix semi- refers to the fact that the homotopy which takes L to the trivial loop can leave U and travel to other parts of X.

The property of semilocal simple connectedness is important because it is a necessary and sufficient condition for a connected, locally pathwise-connected space to have a universal cover.


See also

Covering Space, Homotopy, Fundamental Group, Point-Set Topology, Simply Connected, Universal Cover

This entry contributed by John Renze

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Cite this as:

Renze, John. "Semilocally Simply Connected." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/SemilocallySimplyConnected.html

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