Path-connected

Local connectedness

Similarly, a topological space is said to be locally path-connected if it has a base of path-connected sets. An open subset of a locally path-connected space is connected if and only if it is path-connected. This generalizes the earlier statement about ${\displaystyle \mathbb {R} ^{n}}$ and ${\displaystyle \mathbb {C} ^{n}}$, each of which is locally path-connected. More generally, any topological manifold is locally path-connected.