Finite-to-one mappings and large transfinite dimension

Pol (1996) and Arenas (1996) independently introduced transfinite extensions of finite order of mappings by the use of the length of a partially ordered set and Borstʹs order, respectively. By use of the transfinite order of mappings, Arenas introduced a transfinite dimension O-dim based on the Moritaʹs theorem and proved that every countable-dimensional compact metric space has O-dim. Then he asked whether the converse is true. In the present note, we shall show that both the transfinite extensions given by Pol and Arenas are the same if we ignore the values, and give an affirmative answer to Arenasʹ question as follows: a metrizable space X has the order dimension O-dim X if and only if X has large transfinite dimension Ind X. Furthermore, we shall prove that if a metrizable space X has the order dimension O-dim, then Ind X ⩽ O-dim X and O-dim Sα = α for every ordinal number α < ω1, where Sga is Smirnovʹs compactum.