Convexity preserving mappings in subbase convexity theory

✍ Scribed by J. Van Mill; M. Van De Vel

• Publisher: Elsevier Science
• Year: 1978
• Weight: 700 KB
• Volume: 81
• Category: Article
• ISSN: 1385-7258
• DOI: 10.1016/1385-7258(78)90025-2

✦ Synopsis

The present paper is a continuation of investigations on subbase convexity theory, st arte d in [7] and in [8]. We are now concerned with so-called convexity preserving (cp) mappings, a notion comparable to affine mappings in vector space theory.

A first result is a characterization of cp maps in terms of subbasic line segments, from which it can be deduced that normal binary subbases on a given space are incomparable. It is also proved that a cp map commutes with the fundamental operations on spaces with normal binary subbases. This leads to a uniqueness theorem of induced Jensen mappings on superextensions, and to a new order theoretic classification of the superextensions of a space. We finally prove the existence of metrics which are intimately related to normal binary subbases of metrizable compacta.