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 $R^{n}$ . 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 continuous map $γ : [0, 1] \to X$ . If $γ (0) = x$ and $γ (1) = y$ , then we say that $γ$ is a path from $x$ to $y$ .

Path-connectedness

A topological space $X$ is path-connected if it is non-empty and for all $x, y \in X$ , there exists a path from $x$ to $y$ in $X$ .

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.

Every path-connected space is connected. Proof.

We know that all continuous maps from a path-connected space $X$ to a discrete space $D$ are constant iff $X$ is connected. Let $f$ be an arbitrary continuous map from $X$ to $D$ . Claim: $f$ is constant. Let $x, y \in X$ . There is a path $γ : [0, 1] \to X$ from $x$ to $y$ , and $f \circ γ$ is then a continuous map $[0, 1] \to D$ . But $[0, 1]$ is connected, so $f \circ γ$ is constant, and in particular, $f (x) = f (γ (0)) = f (γ (1)) = f (y)$ , as required.

Proving connectedness

It can be quite hard to prove that a space is connected by relying on the definition of connectedness. For example, how do you show that a disk in $R^{2}$ doesn't have a separation? The implication from path-connectedness to connectedness can be very useful in such situations.

Example

Convex sets

A subset $X$ of $R^{n}$ is convex iff for all $x, y \in X$ and $t \in [0, 1]$ , we have $(1 - t) x + t y \in X$ . For instance, the convex subsets of $R$ are precisely the intervals. Every convex subset of $R^{n}$ is path-connected (with $x, y \in X$ , $t \mapsto (1 - t) x + t y$ is a path from $x$ to $y$ ). Hence, every convex subset of $R^{n}$ is connected.

Geometrically, convex sets are those where every point in the set is connected by a straight line segment.

Context

Source

Leinster p76