# Subset of Toset is Toset

## Theorem

Let $\left({S, \preceq}\right)$ be a totally ordered set.

Let $T \subseteq S$.

Then $\left({T, \preceq \restriction_T}\right)$ is also a totally ordered set.

In the above, $\preceq \restriction_T$ denotes the restriction of $\preceq$ to $T$.

## Proof

As $\left({S, \preceq}\right)$ is a totally ordered set, the relation $\preceq$ is a total ordering, and is by definition:

From Properties of Restriction of Relation, a restriction of a relation which has all those properties inherits them all.

Thus $\preceq \restriction_T$ is also:

and so is also a total ordering.

Hence the result, by definition of totally ordered set.

$\blacksquare$