Cardinal Invariants of Topological Groups

Abstract

AbstractThe definition of the notions of topology and topological space, based on the axiomatic approach, is of necessity of a purely set-theoretic nature. Indeed, a topology is just a family of sets satisfying certain axioms. Not so many elementary and natural properties of sets can be formulated without recourse to special, more complicated, structures or tools of mathematical logic. The most important, and almost the only such property is the cardinality of a set. So no wonder that in General Topology cardinal invariants, that is, characteristics of spaces preseved by homeomorphisms and formulated in terms of cardinal numbers and of families of sets, play a central, almost universal, role. Cardinal invariants measure the size of the space in various ways, the local behaviour of the space, and, most importantly, they are used to bring to light specific features of the space. When we consider continuous mappings, it is important to know which cardinal invariants do not increase under certain natural restrictions on these mappings. When studying products of spaces, it is most useful, whatever our main interest may be, to know how certain cardinal invariants behave under the Tychonoff product operation. Some of the questions of this kind are quite deep and difficult, and the work on them has generated much of the progress not only in General Topology, but in Set Theory and in Mathematical Logic as well. To make the point, it is enough to mention the following question. Suppose that the Souslin number of a space X is countable. Is the Souslin number of the square X x X of X countable? This question, which is so easy to formulate and understand, is intimately related to the famous Souslin Conjecture and to Martin's Axiom, coined in Mathematical Logic, both of which have so many consequences in Topology and Analysis.KeywordsTopological GroupContinuous HomomorphismAbelian Topological GroupCountable NetworkParatopological GroupThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.