\begin{definition}[Definition:Component (Topology)/Definition 1]
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 $y$.
From Connectedness of Points is Equivalence Relation, $\sim$ is an equivalence relation.
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 $x \in S$, then the '''component of $T$ containing $x$''' (that is, the set of points $y \in S$ with $x \sim y$) is denoted by $\map {\operatorname {Comp}_x} T$.
\end{definition}