The open sets in a topological space are those sets A for which A°=>A. Sets for which A 0~°^A -"α-sets"-or A°~^A -"β-sets"-may naturally be considered as more or less "nearly open". In this paper the structure of these sets and classes of sets are investigated, and some applications are given.Topologies determining the same class of α-sets also determine the same class of β-sets, and vice versa. The class of β-sets forms a topology if and only if the original topology is extremally disconnected. The class of α-sets always forms a topology, and topologies generated in this way-"α:-topolgies"-are exactly those where all nowhere dense sets are closed.The class of all topologies which determine the same α-sets is convex in the ordering by inclusion, the ^-topology being its finest member. Most topologies ordinary met with are the coarsest members of their corresponding classes; in particular this is the case for all regular topologies.All topologies determining the same α-sets also determine the same continuous mappings into arbitrary regular spaces.

Let {α 1 , α 2 , . . . } be a sequence of real numbers outside the interval [−1, 1] and µ a positive bounded Borel measure on this interval. We introduce rational functions ϕ n (x) with poles {α 1 , . . . , α n } orthogonal on [−1, 1] and establish some ratio asymptotics for these orthogonal rational functions, i.e. we discuss the convergence of ϕ n+1 (x)/ϕ n (x) as n tends to infinity under certain assumptions on the measure and the location of the poles. From this we derive asymptotic formulas for the recurrence coefficients in the three term recurrence relation satisfied by the orthonormal functions.

This paper surveys the closely related topics included in the title. Emphasis is given to the parallelism between the approach using (Perron-Caratheodory) continued fractions to solve the trigonometric moment problem, and the alternate development that proceeds from the sequence of moments {ti n }™«>> to the linear functional ft, to the Szego polynomials and their reciprocal and associated polynomials, and to the quadrature formula for fi and the solution of the moment problem. CONTENTS

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