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:

  • reflexive
  • antisymmetric
  • transitive
  • connected

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:

  • reflexive
  • antisymmetric
  • transitive
  • connected

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
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