The dependence of the abstract boundary classification on a set of curves I: An algebra of sets on bounded parameter property satisfying sets of curves

Abstract

The abstract boundary uses sets of curves with the bounded parameter property (b.p.p.) to classify the elements of the abstract boundary into regular points, singular points, points at infinity and so on. To study how the classification changes as this set of curves is changed it is necessary to describe the relationships between these sets of curves in a way that reflects the effect of the curves on the classification. The usual algebra of sets fails to do this. We remedy this situation by generalising inclusion, intersection, and union: producing an algebra of sets on the set of all b.p.p. satisfying sets of curves that does appropriately describe the relative effects on the classification. In Part II we use this algebra of sets to show how the classification changes as the set of b.p.p. satisfying set of curves is changed with respect to this generalization.