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 $y$.

Also see

 * Equivalence of Definitions of Component