Definition:Component (Topology)/Definition 3

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 $y$.

The component of $T$ containing $x$ is defined as:
 * the maximal connected subspace of $T$ that contains $x$.

Also see

 * Equivalence of Definitions of Component