Path-Connectedness — Mathematics Knowledge System

Overview

A topological space X is path-connected if any two points x, y ∈ X can be joined by a path — a continuous function γ: [0,1] → X with γ(0) = x and γ(1) = y. This is a stronger, more hands-on notion than plain connectedness: instead of merely forbidding a clean open/open split of the space, path-connectedness demands you can actually walk continuously from any point to any other while staying inside X. Every path-connected space is connected, but the converse fails — the topologist's sine curve is the classic counterexample: connected, yet with no path joining a point on its oscillating part to a point on its limiting vertical segment. Path-connectedness is also the foundation for the fundamental group, since loops (paths starting and ending at the same point) are exactly what π₁ organizes into a group.

Intuition

Connectedness only forbids cutting a space cleanly into two open pieces; path-connectedness asks something much more concrete — can you draw an unbroken line (a continuous path) from any point to any other, never leaving the space? A solid disk is obviously path-connected: draw a straight line between any two points. A space made of two separate blobs is not even connected, let alone path-connected. The subtle case is the topologist's sine curve: the graph of sin(1/x) for 0<x≤1, together with the vertical segment {0}×[-1,1] added at x=0. This set is connected (you cannot separate it into two open pieces), but no continuous path can start on the oscillating curve and end on the vertical segment — a path attempting to do so would have to oscillate infinitely fast near x=0, violating continuity.

Formal Definition

Definition

A path in a topological space X from x to y is a continuous function γ: [0,1] → X with γ(0)=x, γ(1)=y. X is path-connected if:

\forall\, x, y \in X,\ \exists\, \gamma: [0,1] \to X \text{ continuous with } \gamma(0) = x,\ \gamma(1) = y

Path-connectedness

x \sim y \iff \exists \text{ a path from } x \text{ to } y

This relation is reflexive (constant path), symmetric (reverse the path), and transitive (concatenate paths), partitioning X into path-components

Path-component equivalence relation

X \text{ path-connected} \implies X \text{ connected}

The converse fails in general (the topologist's sine curve is the standard counterexample)

One-way implication

Notation

Notation	Meaning
$\gamma: [0,1] \to X$	A path in X — a continuous map from the unit interval
$\gamma * \delta$	Concatenation of two paths (δ starting where γ ends), traversing γ then δ at double speed
$[x]$	The path-component of x — the set of all points reachable from x via some path, i.e. its equivalence class under the path-connectivity relation

Derivation

Deriving that path-connectedness implies connectedness, by contradiction using the Intermediate Value Theorem style argument on the connected interval [0,1].

\text{Suppose } X = U \sqcup V \text{ is a separation (disjoint nonempty open sets, union } X\text{)}

Assume for contradiction that X is disconnected

\text{Pick } x \in U,\ y \in V, \text{ and a path } \gamma:[0,1]\to X \text{ with } \gamma(0)=x,\ \gamma(1)=y

Path-connectedness guarantees such a γ exists

\gamma^{-1}(U) \text{ and } \gamma^{-1}(V) \text{ are open in } [0,1], \text{ disjoint, nonempty, and cover } [0,1]

Continuity of γ pulls back the separation to [0,1]

\text{This is a separation of } [0,1], \text{ contradicting that } [0,1] \text{ is connected.}

Hence no separation of X can exist if X is path-connected, so X must be connected

Proofs

The topologist's sine curve is connected but not path-connected

$\text{Let } S = \{(x, \sin(1/x)) : 0 < x \le 1\} \cup (\{0\}\times[-1,1])$ (Union of the oscillating graph and the added vertical segment at x=0)
$S \text{ is connected: it is the closure of the connected set } \{(x,\sin(1/x)): 0<x\le 1\}, \text{ and the closure of a connected set is connected.}$ (The oscillating piece is the continuous image of the connected interval (0,1], hence connected; its closure adds exactly the vertical segment and stays connected.)
$\text{Suppose a path } \gamma:[0,1]\to S \text{ has } \gamma(0) = (0,0) \text{ and } \gamma(1) = (1,\sin 1).$ (Assume for contradiction that such a path exists, joining the vertical segment to the oscillating part)
$\text{As } t \to t_0^{+} \text{ (the last time } \gamma \text{ touches the vertical segment), the x-coordinate of } \gamma(t) \text{ must pass through every value } 1/(n\pi) \text{ infinitely often in any neighborhood of } t_0, \text{ forcing the y-coordinate to oscillate between } -1 \text{ and } 1 \text{ without settling, contradicting continuity of } \gamma \text{ at } t_0.$ (A continuous path cannot make its y-coordinate oscillate through the full range [-1,1] infinitely often in an arbitrarily small time interval — this violates continuity at t₀)
$\therefore \text{no such path } \gamma \text{ exists, so } S \text{ is not path-connected.}$ (Contradiction shows the assumed path cannot exist)

Properties

Path-connected implies connected

X \text{ path-connected} \implies X \text{ connected}

Condition: Since [0,1] is connected and continuous images of connected sets are connected, any path's image is connected; gluing overlapping connected path-images that share basepoints keeps the whole space connected.

Continuous image of path-connected is path-connected

f: X \to Y \text{ continuous}, X \text{ path-connected} \implies f(X) \text{ path-connected}

Example: Compose any path in X with f to get a path in f(X) between the images of its endpoints.

Open connected subsets of ℝⁿ are path-connected

U \subseteq \mathbb{R}^n \text{ open and connected} \implies U \text{ path-connected}

Condition: A key special case: for open subsets of Euclidean space, connected and path-connected coincide.

Union of path-connected sets sharing a point

\{A_\alpha\} \text{ path-connected}, \bigcap_\alpha A_\alpha \neq \emptyset \implies \bigcup_\alpha A_\alpha \text{ path-connected}

Condition: Concatenate a path within A_α to the common point, then a path within A_β.

Path-components partition X

X = \bigsqcup_i P_i

Condition: Each path-component P_i is path-connected, and distinct path-components are disjoint, since path-connectivity is an equivalence relation.

Applications

Motion planning in configuration space requires path-connectedness of the free (obstacle-free) region to guarantee a robot can actually move from a start to a goal configuration without teleporting.

Worked Examples

For any x, y ∈ C, define the straight-line path.
$\gamma(t) = (1-t)x + ty, \quad t \in [0,1]$
γ is continuous (each coordinate is an affine, hence continuous, function of t), and by convexity of C, γ(t) ∈ C for all t ∈ [0,1].
$\gamma(t) \in C \ \forall t \in [0,1] \text{ (definition of convexity)}$
γ(0) = x and γ(1) = y, so γ is a path in C from x to y.
$\gamma(0)=x,\ \gamma(1)=y$

Answer: Every convex set is path-connected, via the straight-line path between any two points.

Practice Problems

Difficulty 4/10

Path-connectedness relates to connectedness how?

They are always equivalent for any topological spacePath-connected implies connected, but not conversely in generalConnected implies path-connected, but not converselyNeither implies the other

Difficulty 5/10

Is ℝ² \ {(0,0)} (the plane minus the origin) path-connected?

Difficulty 7/10

Prove that an open connected subset U of ℝⁿ is path-connected.

Quiz

The topologist's sine curve demonstrates that:

Connectedness implies path-connectednessPath-connectedness implies connectedness, but not converselyNeither notion implies the otherAll subsets of ℝ² are path-connected

For OPEN subsets of ℝⁿ, connectedness and path-connectedness:

Are unrelatedAlways coincidePath-connectedness always failsConnectedness always fails

Summary

X is path-connected if any two points can be joined by a continuous path γ: [0,1] → X.
Path-connected always implies connected, but not conversely — the topologist's sine curve is connected yet not path-connected.
For open subsets of ℝⁿ, connectedness and path-connectedness coincide, via a ball-chaining argument.
Path-connectivity is an equivalence relation, partitioning any space into path-components; this partition underlies the definition of the fundamental group.

References

WebsiteWikipedia — Path-connected space
WebsiteWikipedia — Topologist's sine curve
BookMunkres, J. Topology, 2nd ed. Ch. 3.