Cardinal functions for compact and metrizable topological spaces

Abstract

Cardinal functions are mappings from the class of topological spaces into the class of infinite cardinal numbers. A systematic study of cardinal functions began in the mid-1960s but most of the fundamental results were obtained long before. Notable researchers such as Alexandroff, Urysohn, Cech, Jones, Hajnal, Juhàsz, Arhangel'skii, and others contributed to the study of cardinal functions. Cardinal functions are very useful to obtain bounds on the cardinality of a topological space X. The importance and study of cardinal functions can also be justified by the fact that such functions where fundamental in solving long standing important problems in topology. We will introduce the basic global and local cardinal functions on a topological space X. Then we will outline some basic inequalities between these cardinal functions. The Pol-Sapirovskii technique will then be applied to obtain some interesting bounds on ƖXƖ. Finally we consider two of the most important classes of topological spaces, namely compact and metrizable spaces. We will see that cardinal functions on compact spaces are very interesting and useful, one only has to mention that weight and net weight coincide for such spaces. Due to the nice qualities of metrizable spaces, inequalities among cardinal functions tend to be dull. However, it is possible to obtain precise information about the cardinality of such spaces.