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The connected component of a point x is the largest connected subset containing x—formally, the union of all connected subsets that contain x. Connected components partition any topological space into maximal connected pieces, and they are always closed sets. This decomposition reveals the global structure of a space: a space is connected if and only if it has exactly one component. In totally disconnected spaces like the Cantor set, every component is a single point. The number and nature of connected components provide a coarse but powerful topological invariant.
Draw examples: identify the components of the real line minus a few points, then of a union of disjoint circles. Move to the topologist's sine curve to see that components can be connected but not path-connected, sharpening the distinction.
Students often assume connected components must be open—they are always closed but not necessarily open. Also, path-components and connected components can differ; path-connectedness is strictly stronger than connectedness.
Given a topological space X, the connected component of a point x is the largest connected subset of X that contains x. Formally, it is the union of all connected subsets of X that contain x. Since any union of connected sets sharing a common point is connected, this union is itself connected and is maximal: no strictly larger connected subset of X contains x. Every point of X belongs to exactly one connected component, and distinct connected components are disjoint, so the connected components partition X into maximal connected pieces.
Connected components are always closed sets. To see why: the closure of a connected set is connected, so if C is the connected component of x, then the closure cl(C) is a connected set containing x. By maximality of C, we must have cl(C) ⊆ C, which means C = cl(C) — the component is closed. However, connected components are not necessarily open. In the rational numbers ℚ with the subspace topology from ℝ, every connected component is a single point (since between any two rationals lies an irrational, allowing a disconnection). These singletons are closed but not open in ℚ. Components are guaranteed to be open only when the space is locally connected — a separate condition beyond basic connectedness.
The component decomposition reveals the global structure of a space at a coarse level. A space is connected if and only if it has exactly one component. The real line minus two points, ℝ \ {0, 1}, has three components: (−∞, 0), (0, 1), and (1, ∞). At the other extreme, a totally disconnected space like the Cantor set has every component equal to a single point — the space is maximally fragmented. The number and nature of connected components form a topological invariant: homeomorphic spaces have the same component structure.
It is important to distinguish connected components from path-components. The path-component of x is the set of all points that can be joined to x by a continuous path. Every path-component is contained in a connected component, and in most familiar spaces (manifolds, CW complexes, locally path-connected spaces) the two notions coincide. But they can diverge: the topologist's sine curve — the closure of the graph of sin(1/x) for x > 0 — is connected (one connected component) but has two path-components, because no continuous path can cross the infinitely oscillating accumulation at the y-axis. This distinction matters in algebraic topology, where path-connectedness rather than connectedness is typically the relevant condition.