Cardinal functions for linearly ordered topological spaces

Abstract

Cardinal functions extend such important properties as countable base, separable, and first countable to higher cardinality. Cardinal functions then allow one to formulate, generalise, and prove related results in a systematic and elegant manner. In addition, cardinal functions allow one to make precise quantitative comparisons between certain topological properties. Experience indicates that the idea of a cardinal function is one of the most useful and important unifying concepts in all of set-theoretic topology. A cardinal function is a function ϕ from the class of all topological spaces (or some precisely defined subclass) into the class of all infinite cardinals such that ϕ(X) = ϕ (Y) whenever X and Y are homeomorphic. An obvious example of a cardinal function is cardinality, denoted ƖXƖ and equal to the number of points in X plus ω. Perhaps the most useful cardinal function is weight, defined by

ω (X) = min{ƖB Ɩ : B a base for X} + ω .

In the thesis, we first define carefully the most important cardinal functions in general topological spaces and study the relationships between them. Important results related to bounds on jXj in terms of other cardinal functions are then investigated. In particular, difficult inequalities like jXj _ 2L(X)__(X) for X 2 T2, jXj ≤ 2c(X)__(X) for X 2 T2, jXj _ 2s(X)_ (X) for X 2 T1, and jXj ≤ 22s(X) for X 2 T2, are studied. Finally, and most importantly, the behaviour of the above mentioned cardinal functions for the specific class of linearly ordered topological spaces is investigated.