Connected component - Knowlpedia

Empty set and whole space: ∅ , X ∈ T \varnothing,X\in\mathcal T ∅ , X ∈ T .
Arbitrary unions: the union of any collection of members of T \mathcal T T belongs to T \mathcal T T .
Finite intersections: the intersection of finitely many members of T \mathcal T T belongs to T \mathcal T T .

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

C(x)=\bigcup\{\,C\subseteq X : C \text{ is connected and } x\in C\,\},

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

Remarks

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.