Path connectedness. Definition
Viewing topology from the lens of Euclidean space suggests a variant of the notion of connectedness: path-connectedness.
The Euclidean lens
A topological space is said to be an n-dimensional manifold if it is Hausdorff and has an open cover by subsets each homeomorphic to an open ball in
Paths
Let
A path in
Path-connectedness
A topological space
The relevance to standard connectedness is quickly apparent:
Every path-connected space is connected.
Proof on reverse side. Note that the converse is false, and thus, path-connectedness is a stronger condition than vanilla conectedness.