Connected component

A connected component of a topological space $X$ is a maximal connected set (with respect to inclusion). Concretely, for a point $x\in X$, its connected component is

and $C(x)$ is the largest connected subset of $X$ containing $x$.

Connected components give a canonical decomposition of $X$ into connected pieces; compare this with path-connectedness , which uses paths .

Examples:

• In $\mathbb{R}\setminus\{0\}$ (usual topology), the connected components are $(-\infty,0)$ and $(0,\infty)$.
• In a discrete topological space, every singleton $\{x\}$ is a connected component.