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

Theorem
Let $S$ be a set.

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

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

Let $\mathcal R \restriction_T \, \subseteq T \times T$ be the restriction of $\mathcal R$ to $T$.

Then $\mathcal R \restriction_T$ is not necessarily a non-connected relation on $T$.

Proof
Proof by Counterexample:

Let $S = \left\{{a, b}\right\}$.

Let $\mathcal R = \left\{{\left({a, a}\right), \left({b, b}\right)}\right\}$.

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

Now let $T = \left\{{a}\right\}$.

Then $\mathcal R \restriction_T \, = \left\{{\left({a, a}\right)}\right\}$.

Then $\mathcal R \restriction_T$ is trivially connected on $T$.

Also see

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