Several Problems of Continua Theory

Abstract

This chapter discusses some problems related to the concept of the confluency of mappings. The problems discussed are whether confluent mappings preserve rational continua and whether weakly confluent mappings preserve Class A. Both monotone mappings and open mappings are confluent, and it is rather easy to prove that they do preserve rational continua. However, in general, first problem seems to be more involved than in the case of monotone or open mappings. What probably causes the difficulty is the fact that a slightly larger class of continua, namely that of Suslinian ones, is preserved by confluent mappings and too many examples of nonrational Suslinian continua are not known. A continuum is Suslinian provided that each collection of mutually disjoint non-degenerate subcontinua of it is countable. Weakly confluent mappings preserve the class of Suslinian continua; however, they do not necessarily preserve rational continua.