Definition:Component (Topology)

Let $$T$$ be a topological space.

Let us define an equivalence relation $$\sim $$ on $$T$$ as follows:

$$x \sim y$$ iff there exists a connected subspace of $$T$$ that contains both $$x$$ and $$y$$.

It is clear that $$\sim $$ is an equivalence: reflexivity and symmetry are obvious, and transitivity follows from Spaces with Connected Intersection have Connected Union.

The resulting equivalence classes are called the (connected) components of $$T$$.

If $$x\in T$$, then the component of $$T$$ containing $$x$$ (that is, the set of points $$y\in T$$ with $$x \sim y$$) is denoted by $$ \operatorname{Comp}_x(T).$$

Alternative Definitions
The component $$C$$ of $$T$$ containing $$x$$ can be alternatively defined as:

(Here, "maximal" is used in the sense that all connected subspaces of $$T$$ are themselves subsets of some component of $$T$$.)
 * $$C = \bigcup\{A\subseteq T: A\text{ is connected and contains } x\}$$;
 * the maximal connected subspace of $$T$$ that contains $$x$$.

The fact that these definitions are equivalent is demonstrated in Equivalence of Definitions of Component.