Connectedness

Three equivalent definitions of "connected"

A topological space X is connected if any one of the following holds (the three are equivalent):

1. X cannot be written as a disjoint union of two non-empty open subsets — there is no decomposition X = U ∪ V with U, V open, non-empty, and U ∩ V = ∅.
2. The only subsets of X that are both open and closed (clopen) are ∅ and X.
3. Every continuous map f: X → {0, 1} (with the discrete topology on the codomain) is constant.

Equivalence (1) ↔ (2): a non-trivial clopen subset U gives the decomposition X = U ∪ (X \ U). Equivalence (2) ↔ (3): a continuous map to {0, 1} has clopen pre-images, so it must be constant if the only clopen sets are trivial.

Path-connectedness

X is path-connected if for every pair of points a, b ∈ X there is a continuous map γ: [0, 1] → X with γ(0) = a and γ(1) = b. The image of γ is called a path from a to b.

Path-connectedness defines an equivalence relation on X: a ~ b if there is a path between them. Equivalence classes are path components. π₀(X) is the set of path components; for a path-connected space, |π₀(X)| = 1.

Path-connected ⇒ connected

Suppose X is path-connected. To show X is connected, assume X = U ∪ V is a disconnection. Pick a ∈ U, b ∈ V. There's a path γ from a to b. Then γ⁻¹(U) and γ⁻¹(V) are disjoint, open, and non-empty subsets of [0, 1] whose union is [0, 1] — contradicting the connectedness of [0, 1]. So X is connected. ∎

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

Let G = {(x, sin(1/x)) : 0 < x ≤ 1} (the "wiggly graph") and S = {0} × [−1, 1] (the limit segment on the y-axis). The topologist's sine curve is T = G ∪ S.

T is connected: G is the continuous image of (0, 1] under a continuous map, so G is connected. S is the closure of G in R² minus G, so adjoining S still produces a connected set (the closure of a connected set is connected, and T equals the closure of G in R²).

T is not path-connected: suppose γ: [0, 1] → T is a path with γ(0) = (0, 0) ∈ S and γ(1) = (1, sin(1)) ∈ G. Let t* be the supremum of t with γ(t) ∈ S. By continuity γ(t*) ∈ S. For any t > t*, γ(t) ∈ G. But then γ(t).x → 0⁺ as t → t*, so γ(t).y = sin(1/γ(t).x) takes every value in [−1, 1] infinitely often — γ cannot have a limit at t*, contradicting continuity.

So T is connected but not path-connected. The phenomenon is purely local: at the segment S, T fails to be locally path-connected, and that local failure breaks the equivalence with global connectedness.

Connected and path components

The connected component C(x) of a point x ∈ X is the largest connected subset of X containing x. The path component P(x) of x is the largest path-connected subset of X containing x.

Always: P(x) ⊆ C(x). For the topologist's sine curve, the two components of (0, 0) differ — P(0, 0) = S, but C(0, 0) = T.

Connected components are always closed in X. Path components are not always closed (the wiggly part of T is the path component of (1, sin 1) but its closure includes the y-axis segment, which is in a different path component).

A reference table

Space	Connected?	Path-connected?	Notes
Single point	Yes	Yes	Trivially both
Open interval (0, 1)	Yes	Yes	Any interval works
Two disjoint disks in R²	No	No	Two connected components
Q (rationals) ⊂ R	No	No	Totally disconnected — components are singletons
Closed disk D²	Yes	Yes	Convex sets in R^n are always path-connected
Topologist's sine curve T	Yes	No	The classical counter-example
GL(n, R) for n ≥ 1	No (n=1) / 2 comps (n≥1)	2 path components	det > 0 and det < 0 pieces
S^n for n ≥ 1	Yes	Yes	Connected manifolds are path-connected (locally PC)

Historical context

The intuitive idea of "one piece" appears throughout 19th-century analysis, but Camille Jordan gave the first formal definition in his 1893 Cours d'analyse: a set is connected if it cannot be partitioned into two parts each containing none of the limit points of the other. Felix Hausdorff's 1914 Grundzüge der Mengenlehre introduced general topological spaces and refined the definition to the open-set version used today.

The Polish school — Sierpiński, Kuratowski, Mazurkiewicz — explored pathological examples in the 1910s and 1920s. Knaster and Kuratowski's "broom-like" connected-not-path-connected spaces appeared in 1921. The topologist's sine curve became the iconic example by the mid-20th century, appearing in every general-topology textbook from Kelley (1955) onwards.

Hahn–Mazurkiewicz (1914) proved that a Hausdorff space is the continuous image of [0, 1] if and only if it is compact, connected, locally connected, and second-countable — characterising "Peano continua." This was a 30-year-old open problem.

Why connectedness is everywhere

• Intermediate value theorem. The continuous image of a connected space is connected. Connected subsets of R are intervals. So a continuous f: [a, b] → R changing sign must hit zero — IVT in one line.
• Identity theorem. Two holomorphic functions agreeing on a set with a limit point in a connected domain agree everywhere. Without connectedness this fails — different formulas on different components.
• Numerical solvers — bisection. The bisection method for f(x) = 0 relies on IVT, hence connectedness of [a, b]. Each step halves a connected sub-interval.
• Computer graphics — flood fill. Filling a pixel region with a colour traverses the connected component of the starting pixel. Algorithms (BFS, scanline fill) compute path components in the grid graph.
• Riemann surfaces. Two holomorphic structures on a connected Riemann surface agree on the entire surface if they agree on any open subset — analytic continuation hinges on connectedness.
• Manifold classification. The classification of closed connected surfaces (Möbius–Kerékjártó, 1923) requires "connected" as a hypothesis — otherwise the classification is just "do it on each component."
• Galois theory. The Galois group of a connected scheme acts as the étale fundamental group π₁^ét — a number-theoretic analogue of topological connectedness.

JavaScript — flood fill computes a connected component

// Connected components on a pixel grid (4-connectivity)
function connectedComponent(grid, startR, startC) {
  const rows = grid.length, cols = grid[0].length;
  const target = grid[startR][startC];
  const visited = Array.from({ length: rows }, () => Array(cols).fill(false));
  const stack = [[startR, startC]];
  const component = [];

  while (stack.length) {
    const [r, c] = stack.pop();
    if (r < 0 || c < 0 || r >= rows || c >= cols) continue;
    if (visited[r][c] || grid[r][c] !== target) continue;
    visited[r][c] = true;
    component.push([r, c]);
    stack.push([r+1, c], [r-1, c], [r, c+1], [r, c-1]);
  }
  return component;
}

// Two disjoint blobs of 1s — two connected components
const grid = [
  [1, 1, 0, 0, 0],
  [1, 0, 0, 1, 1],
  [0, 0, 0, 1, 1],
  [0, 0, 0, 0, 0],
];
console.log(connectedComponent(grid, 0, 0).length);  // 3 cells — top-left blob
console.log(connectedComponent(grid, 1, 3).length);  // 4 cells — bottom-right blob

// Check connectedness of a subset of R via interval-property
function isInterval(sortedPoints) {
  // For a finite set of points to be 'connected' in R, they must be a single interval
  for (let i = 1; i < sortedPoints.length; i++) {
    if (sortedPoints[i] - sortedPoints[i-1] > 1e-9 * Math.max(1, sortedPoints[i])) return false;
  }
  return true;
}
// The rationals in [0,1] sampled at 100 points are densely connected — but rationals
// themselves are totally disconnected: every rational is its own connected component.

Variants and stronger conditions

• Locally connected. Every point has arbitrarily small connected neighbourhoods. Implies connected components are open. R^n and manifolds are locally connected; rationals are not.
• Locally path-connected. Every point has arbitrarily small path-connected neighbourhoods. Combined with connectedness, implies path-connectedness. Manifolds satisfy this.
• Simply connected. Path-connected plus trivial fundamental group: every loop contracts to a point. Disk is simply connected; annulus is not.
• Totally disconnected. Every connected component is a single point. Examples: rationals, Cantor set, p-adic numbers Q_p.
• n-connected. π_k(X) trivial for k ≤ n. 0-connected = path-connected; 1-connected = simply connected. The full hierarchy of "higher-dimensional connectedness."
• Hyperconnected. Any two non-empty open sets intersect. The Zariski topology on irreducible varieties is hyperconnected.

Common misconceptions

• "Connected and path-connected are the same." They agree on open subsets of R^n and on manifolds (locally path-connected spaces). They disagree on the topologist's sine curve, Warsaw circle, comb space, and many other classical examples.
• "Disconnected means physically separated." Not necessarily. The rationals Q sit densely inside R but are totally disconnected — every rational is its own component. The disconnection is topological, not visual.
• "Connectedness is preserved under set intersection." Two connected sets can intersect in a disconnected set — e.g. two overlapping annuli in R². The union of two connected sets is connected only if they intersect.
• "A finite space can't be connected unless it's a single point." Wrong — the Sierpiński space {0, 1} with open sets {∅, {1}, {0, 1}} is connected and has two points. Finite spaces with non-trivial topologies can be highly non-trivial.
• "Discrete spaces are disconnected." True for spaces with ≥ 2 points (every singleton is clopen). The empty space and a single point are connected by convention or by vacuous reasoning.
• "Connectedness implies compactness." No — R is connected and not compact. Compactness and connectedness are independent properties. A space can have any combination of (connected? compact?).