Topology of Function Spaces

by Marsh, Andrew James

This dissertation is a study of the relationship between a topological space X and various higherorder objects that we can associate with X. In particular the focus is on C(X), the set of all continuous realvalued functions on X endowed with the topology of pointwise convergence, the compactopen topology and an admissible topology. The topological properties of continuous function universals and zero set universals are also examined. The topological properties studied can be divided into three types (i) compactness type properties, (ii) chain conditions and (iii) sequential type properties. The dissertation begins with some general results on universals describing methods of constructing universals. The compactness type properties of universals are investigated and it is shown that the class of metric spaces can be characterised as those with a zero set universal parametrised by a sigma-compact space. It is shown that for a space to have a Lindelof-Sigma zero set universal the space must have a sigma-disjoint basis. A study of chain conditions in Ck(X) and Cp(X) is undertaken, giving necessary and sufficient conditions on a space X such that Cp(X) has calibre (kappa,lambda,mu), with a similar result obtained for the Ck(X) case. Extending known results on compact spaces it is shown that if a space X is omegabounded and Ck(X) has the countable chain condition then X must be metric. The classic problem of the productivity of the countable chain condition is investigated in the Ck setting and it is demonstrated that this property is productive if the underlying space is zerodimensional. Sufficient conditions are given for a space to have a continuous function universal parametrised by a separable space, ccc space or space with calibre omega1. An investigation of the sequential separability of function spaces and products is undertaken. The main results include a complete characterisation of those spaces such that Cp(X) is sequentially separable and a characterisation of those spaces such that Cp(X) is strongly sequentially separable.