Restriction of Strict Total Ordering is Strict Total Ordering

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Theorem

Let $\struct {S, \prec}$ be a strict total ordering.

Let $T \subseteq S$.

Let $\prec \restriction_T$ be the restriction of $\prec$ to $T$.

Then $\prec \restriction_T$ is a strict total ordering of $T$.

Proof

By definition of strict total ordering, $\prec$ is:

$(1): \quad$ a relation which is transitive and antireflexive

$(2): \quad$ a relation which is connected.

By Reflexive Reduction of Transitive Antisymmetric Relation is Strict Ordering:

$\prec \restriction_T$ is a strict ordering.

It follows from Restriction of Connected Relation is Connected that:

$\prec \restriction_T$ is connected.

Thus $\prec \restriction_T$ is a strict ordering which is connected.

So by definition $\prec \restriction_T$ is a strict total ordering of $T$.

$\blacksquare$

Sources

• 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 3$: Relations: Exercise $3.7$