Methods of comparison of families of real functions in porosity terms

We consider some families of real functions endowed with the metric of uniform convergence. In the main results of our work we present two methods of comparison of families of real functions in porosity terms. The first method is very general and may be applied to any family of real functions. The second one is more convenient but can be used only in the case of path continuous functions. We apply the obtained results to compare in terms of porosity the following families of functions: continuous, absolutely continuous, Baire one, Darboux, also functions of bounded variation and porouscontinuous, ρ-upper continuous, ρ-lower continuous functions.

Ideals in B1(X) and residue class rings of B1(X) modulo an ideal

<p>This paper explores the duality between ideals of the ring B₁(X) of all real valued Baire one functions on a topological space X and typical families of zero sets, called Z_B-filters, on X. As a natural outcome of this study, it is observed that B₁(X) is a Gelfand ring but non-Noetherian in general. Introducing fixed and free maximal ideals in the context of B₁(X), complete descriptions of the fixed maximal ideals of both B₁(X) and B₁^* (X) are obtained. Though free maximal ideals of B₁(X) and those of B₁^* (X) do not show any relationship in general, their counterparts, i.e., the fixed maximal ideals obey natural relations. It is proved here that for a perfectly normal T₁ space X, free maximal ideals of B₁(X) are determined by a typical class of Baire one functions. In the concluding part of this paper, we study residue class ring of B₁(X) modulo an ideal, with special emphasize on real and hyper real maximal ideals of B₁(X).</p>

On the sets of fixed points of bounded Baire one functions

In this paper, we present some results on typical properties of the sets of fixed points of bounded Baire one functions. In particular, we show that typical elements of a uniformly closed subclass [Formula: see text] of such class of functions have nowhere dense set of fixed points. We also show that typical elements of the class of bounded Baire one functions have [Formula: see text], where [Formula: see text] is an arbitrary continuous Borel measure on the unit interval.

On rings of Baire one functions

<p>This paper introduces the ring of all real valued Baire one functions, denoted by B₁(X) and also the ring of all real valued bounded Baire one functions, denoted by B^∗₁(X). Though the resemblance between C(X) and B₁(X) is the focal theme of this paper, it is observed that unlike C(X) and C^∗(X) (real valued bounded continuous functions), B^∗₁ (X) is a proper subclass of B₁(X) in almost every non-trivial situation. Introducing B₁-embedding and B^∗₁-embedding, several analogous results, especially, an analogue of Urysohn’s extension theorem is established.</p>