\begin{definition}[Definition:Component (Topology)/Definition 3]
Let $T = \struct {S, \tau}$ be a topological space.
Let the relation $\sim $ be defined on $T$ as follows:
:$x \sim y$ {{iff}} $x$ and $y$ are connected in $T$.
That is, {{iff}} there exists a connected set of $T$ that contains both $x$ and $y$.
The '''component of $T$ containing $x$''' is defined as:
: the maximal connected set of $T$ that contains $x$.
\end{definition}