Locally simply connected space

In mathematics, a locally simply connected space is a topological space that admits a basis of simply connected sets.

The cone on the Hawaiian earring is contractible and therefore simply connected, but still not locally simply connected.

All topological manifolds and CW complexes are locally simply connected.

In fact, these satisfy the much stronger property of being locally contractible.

A strictly weaker condition is that of being semi-locally simply connected.