Locally connected space

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

Definition

A topological space $X$ is termed a locally connected space if, for every point $x\in X$, and every open subset $U$ of $X$ containing $x$, there exists an open subset $V$ of $X$ such that $x\in V$, ${\overline {V}}\subseteq U$, and $V$ is a connected space with the subspace topology.

Relation with other properties

Related properties

• Connected space: Being connected does not imply being locally connected, and being locally connected does not imply being connected. Further information: connected not implies locally connected, locally connected not implies connected