nLab locally path-connected space

Context

Topology

topology

algebraic topology

Contents

Definition

A topological space is called locally path-connected if it has a basis of path-connected neighbourhoods. In other words, if for every point $x$ and neighbourhood $V \ni x$, there exists a path-connected neighbourhood $U \subset V$ that contains $x$.

Properties

A locally path-connected space is connected if and only if it is path-connected. In any case, the connected components of a locally path-connected space are the same as its path-connected components.

The condition is a necessary assumption in the

in the form of the condition for

Revised on January 24, 2017 08:36:39 by Urs Schreiber (84.63.65.197)