Math and science::Topology

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.