# 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 LocallyConnected 2013-03-22 12:38:48 2013-03-22 12:38:48 djao (24) djao (24) 5 djao (24) Definition msc 54D05 ConnectedSet ConnectedSpace PathConnected SemilocallySimplyConnected locally path connected