Finite-to-one mappings and large transfinite dimension
by
Yasunao Hattori
Department of Mathematics, Shimane University
Coauthors: Kohzo Yamada

Quite recently, R. Pol introduced a transfinite extension of the order of finite-to-one mappings and investigated the behavior of weakly infinite-dimensional compacta under a continuous mapping with the transfinite order.

Independently, F. G. Arenas gave another transfinite extension of the order of finite-to-one mappings by use of the Borst's order. Then, he extended the covering dimension to transfinite dimension Ø-dim based on the Morita's theorem and proved that every countable-dimensional compact metric space has Ø-dim. He asked whether if every compact metric space having Ø-dim is countable-dimensional. In the present talk, we show that the both of transfinite extensions given by Pol and Arenas are the same if we ignore the values and they are closely related to have large transfinite dimension. Then, we show the following which answers the question above:

Theorem 1. A metrizable space X has the order dimension \romØ-dim if and only if X has large transfinite dimension \rom\Ind. Furthermore, if X has the order dimension \romØ-dim X, then the inequality \rom\Ind X <= \romØ-dim X holds. We do not know the values of \romØ-dim X and \rom\Ind X coincide.

We also have the following.

Theorem 2. Ø-dim S_\alpha = \alpha for every ordinal number \alpha < \omega₁, where S_\alpha is the Smirnov's compactum.

Date received: February 4, 1998

Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caas-26.