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 \( \mathbb{R}^n \). Typical examples of 2-dimensional manifolds (surfaces) are the sphere, the torus and the Klein bottle. Manifolds are enormously important.
Paths
Let \( X \) be a topological space.
A path in \( X \) is a [something]. If \( \gamma(0) = x \) and \( \gamma(1) = y \), then we say that [something] is a path from \( x \) to \( y \).
Path-connectedness
A topological space \( X \) is path-connected if it is non-empty and [...].
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.