A gauge approach to an ordinal index of Baire one functions
نویسندگان
چکیده
We develop a calculus for the oscillation index of Baire one functions using gauges analogous to the moduli of continuity.
منابع مشابه
Extension of Functions with Small Oscillation
A classical theorem of Kuratowski says that every Baire one function on a Gδ subspace of a Polish (= separable completely metrizable) space X can be extended to a Baire one function on X. Kechris and Louveau introduced a finer gradation of Baire one functions into small Baire classes. A Baire one function f is assigned into a class in this heirarchy depending on its oscillation index β(f). We p...
متن کاملar X iv : m at h / 92 02 20 4 v 1 [ m at h . FA ] 2 8 Fe b 19 92 COMPLEXITY OF WEAKLY NULL SEQUENCES
We introduce an ordinal index which measures the complexity of a weakly null sequence, and show that a construction due to J. Schreier can be iterated to produce for each α < ω1, a weakly null sequence (xαn)n in C ( ωω α ) with complexity α. As in the Schreier example each of these is a sequence of indicator functions which is a suppression-1 unconditional basic sequence. These sequences are us...
متن کاملCharacterization of Order Types of Pointwise Linearly Ordered Families of Baire Class 1 Functions
In the 1970s M. Laczkovich posed the following problem: Let B1(X) denote the set of Baire class 1 functions defined on a Polish space X equipped with the pointwise ordering. Characterize the order types of the linearly ordered subsets of B1(X). The main result of the present paper is a complete solution to this problem. We prove that a linear order is isomorphic to a linearly ordered family of ...
متن کاملCodings of separable compact subsets of the first Baire class
Let X be a Polish space and K a separable compact subset of the first Baire class on X. For every sequence f = (fn)n dense in K, the descriptive set-theoretic properties of the set Lf = {L ∈ [N] : (fn)n∈L is pointwise convergent} are analyzed. It is shown that if K is not first countable, then Lf is Π 1 1 complete. This can also happen even if K is a pre-metric compactum of degree at most two, ...
متن کاملThe Szlenk Index and the Fixed Point Property under Renorming
Assume that X is a Banach space such that its Szlenk index Sz X is less than or equal to the first infinite ordinal ω. We prove that X can be renormed in such a way that X with the resultant norm satisfies R X < 2, where R · is the Garcı́a-Falset coefficient. This leads us to prove that if X is a Banach space which can be continuously embedded in a Banach space Y with Sz Y ≤ ω, then, X can be re...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2008