Locally simply 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 locally simply connectedif it satisfies the following equivalent conditions:

1. For every point $x\in X$, and every open subset $V$ of $X$ containing $x$, there is an open subset $U$ of $X$ contained in $V$, and which is simply connected in the subspace topology from $X$.
2. $X$ has a basis of open subsets each of which is a simply connected space with the subspace topology.

Relation with other properties

Stronger properties

• Locally contractible space

Weaker properties

• Semilocally simply connected space
• Locally path-connected space
• Connected space