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

Equivalent definitions in tabular format

No.	Shorthand	A topological space $X$ is termed locally connected if ...
1	locally connected at every point	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 $x\in V$, $V\subseteq U$, and $V$ is a connected space with the subspace topology.
2	weakly locally connected at every point	for every point $x\in X$, and every open subset $U$ of $X$ containing $x$, there exists a subset $A$ of $X$ such that $x$ is in the interon of $A$, $A\subseteq U$, and $A$ is a connected space with the subspace topology.
3	basis of open connected subsets	$X$ has a basis (of open subsets) such that all members of the basis are connected in 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