Definition:Component (Topology)

Definition
Let $T$ be a topological space.

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

$x \sim y$ iff $x$ and $y$ are connected in $T$.

That is, iff there exists a connected subspace of $T$ that contains both $x$ and $y$.

We have that $\sim $ is an equivalence relation, so from the Fundamental Theorem on Equivalence Relations, the points in $T$ can be partitioned into equivalence classes.

These 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 $\operatorname{Comp}_x \left({T}\right)$.

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 $T$.)
 * $C = \bigcup \left\{{A \subseteq T: A \text{ is connected and contains } x}\right\}$
 * the maximal connected subspace of $T$ that contains $x$.

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