Compact subsets of the first Baire class

Abstract

Perhaps the earliest results about pointwise compact sets of Baire class-1 functions are the two selection theorems of E. Helly found in most of the standard texts on real variable (see, e.g., [Lo], [N]). These two theorems are really theorems about a particular example of a compact set of Baire class-1 functions known today as Helly space, the space of all nondecreasing functions from the unit interval I = [0, 1] into itself. More recently, the notion of Baire class-1 function turned out to also be important in some areas of functional analysis (see [R3]). For example, Odell and Rosenthal [OR] showed that the double dual of a separable Banach space E with the weak * topology consists only of Baire class-1 functions defined on the unit ball of E * if and only if the space E contains no subspace isomorphic to 1 . This resulted in a renewed interest in this class of spaces. For example, building on the work of Rosenthal [R2], Bourgain, Fremlin and Talagrand [BFT] proved analogues of the two theorems of Helly for the whole first Baire class. Using their results Godefroy [Go] showed that this class of spaces enjoys some interesting permanence properties. For example, if a compact space K is representable as a compact set of Baire class-1 functions, then so is P (K), the space of all Radon probability measures on K with the weak * topology. Some further permanence properties of this class of spaces were obtained by Marciszewski ([M1], [M2]) and an excellent survey of the early results is given by R. Pol [Po2]. Our paper is an attempt towards a fine structure theory of compact subsets of first Baire class. The first result that we give is a positive answer to a natural question one usually asks in such a context. Theorem 1. Every compact subset of first Baire class contains a dense metrizable subspace.An interesting feature of this result and its proof is that it necessarily requires methods quite different from the classical ones which go through Namioka's analysis of joint versus separate continuity (see [Na]) and which therefore usually produce dense completely metrizable subspaces. Existence of a dense metrizable subspace of K means that K contains many G δ points, a fact first established by Bourgain [Bo]. Note that from the fact that every compact set K of first Baire class contains a G δ point one can conclude that every such K contains a set of G δ points which is everywhere of second category in K. Such a generous conclusion cannot be given for dense metrizable subspaces, revealing the striking difference between the two results. For example, there exists a K of the first Baire class such that every metrizable subspace of K is countable. One example of such K is the split interval S(I), the lexicographically ordered product I × 2 of the unit interval I = [0, 1] and the

Tree structures associated to a family of functions

“…This definition is crucial for the proof of Rosenthal's £' theorem (see [6] or [12]). PROPOSITION 8.…”

“…H REMARK 4. The sequence (/"/ ) k resulting from Proposition 8 can be used to derive the following two well-known results (see [12]).…”

“…The properties of the tree ST ((/")", a, b) can also be used to derive the following well-known dichotomy (see [12]). THEOREM 9.…”

Compact Spaces of the First Baire Class That Have Open Finite Degree

“…We refer to the survey [7] for more information on the subject. Three critical examples of non-metrizable Rosenthal compacta were identified in [14]: the split interval (also known as double arrow space), the Alexandroff duplicate of the Cantor set and the one-point compactification of the discrete set of size continuum. The two latter spaces are not separable, but there is a natural way to supplement them with countably many isolated points so that we obtain three separable nometrizable Rosenthal compacta that form a basis: Theorem 1.…”

Ramsey-theoretic analysis of the conditional structure of weakly-null sequences

“…[44]). Suppose K is a separable compact set of Baire class-1 functions defined on some Polish space X.…”