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

$${\pi }_1(U,x)⟶{\pi }_1(X,x)$$

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