Maximizing an interval order on compact subsets of its domain

Maximal elements of a binary relation on compact subsets of a metric space define a choice function. An infinite extension of transitivity is necessary and sufficient for such a choice function to be nonempty-valued and path independent (or satisfy the outcast axiom). An infinite extension of acyclicity is necessary and sufficient for the choice function to have nonempty values provided the underlying relation is an interval order.