Locally simply connected space

Definition
A topological space $$X$$ is termed locally simply connected if 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.

Stronger properties

 * Locally contractible space

Weaker properties

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