locally connected

A topological space $X$ is locally connected at a point $x\in X$ if every neighborhood $U$ of $x$ contains a connected neighborhood $V$ of $x$. The space $X$ is locally connected if it is locally connected at every point $x\in X$.

A topological space $X$ is locally path connected at a point $x\in X$ if every neighborhood $U$ of $x$ contains a path connected neighborhood $V$ of $x$. The space $X$ is locally path connected if it is locally path connected at every point $x\in X$.

Title	locally connected
Canonical name	LocallyConnected
Date of creation	2013-03-22 12:38:48
Last modified on	2013-03-22 12:38:48
Owner	djao (24)
Last modified by	djao (24)
Numerical id	5
Author	djao (24)
Entry type	Definition
Classification	msc 54D05
Related topic	ConnectedSet
Related topic	ConnectedSpace
Related topic	PathConnected
Related topic	SemilocallySimplyConnected
Defines	locally path connected