# semilocally simply connected

A topological space $X$ is semilocally simply connected if, for every point $x\in X$, there exists a neighborhood $U$ of $x$ such that the map of fundamental groups

 $\pi_{1}(U,x)\longrightarrow\pi_{1}(X,x)$

induced by the inclusion map $U\hookrightarrow X$ is the trivial homomorphism.

A topological space $X$ is connected, locally path connected, and semilocally simply connected if and only if it has a universal cover.

 Title semilocally simply connected Canonical name SemilocallySimplyConnected Date of creation 2013-03-22 12:38:46 Last modified on 2013-03-22 12:38:46 Owner djao (24) Last modified by djao (24) Numerical id 6 Author djao (24) Entry type Definition Classification msc 54D05 Classification msc 57M10 Synonym semilocally 1-connected Synonym locally relatively simply connected Related topic Connected2 Related topic SimplyConnected Related topic ConnectedSpace Related topic LocallyConnected