# 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.