# Subset of Toset is Toset

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## 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.

$\blacksquare$

## Sources

- 1968: A.N. Kolmogorov and S.V. Fomin:
*Introductory Real Analysis*... (previous) ... (next): $\S 3.3$: Ordered sets. Order types