I do understand every subset of a totally ordered set must be a total order as each a, b in the totally ordered set follows either aRb or bRa. I don’t understand what the phrase “for the restriction of the order on X” means? Can anyone explain.
An order in a set X is a relation between elements of X, this is, a subset R of the Cartesian product X × X that satisfies the axioms of order (or total order, etc.).
If Y is a subset of X, the restriction to Y of the order on X is the intersection R ∩ Y×Y of the relation R with the subset Y×Y ⊆ X×X.
In other words, is the same order, but restricted to the subset Y.