Restriction of Non-Connected Relation is Not Necessarily Non-Connected

Theorem
Let $S$ be a set.

Let $\RR \subseteq S \times S$ be a non-connected relation on $S$.

Let $T \subseteq S$ be a subset of $S$.

Let $\RR {\restriction_T} \subseteq T \times T$ be the restriction of $\RR$ to $T$.

Then $\RR {\restriction_T}$ is not necessarily a non-connected relation on $T$.

Proof
Proof by Counterexample:

Let $S = \set {a, b}$.

Let $\RR = \set {\tuple {a, a}, \tuple {b, b} }$.

$\RR$ is a non-connected relation, as can be seen by definition: neither $a \mathrel \RR b$ nor $b \mathrel \RR a$.

Now let $T = \set a$.

Then $\RR {\restriction_T} = \set {\tuple {a, a} }$.

Then $\RR {\restriction_T}$ is trivially connected on $T$.

Also see

 * Properties of Relation Not Preserved by Restriction‎ for other similar results.