2 Sequences of Baire Class 1 Functions 3
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A New Characterization of Baire Class 1 Functions
We give a new characterization of the Baire class 1 functions (defined on an ultrametric space) by proving that they are exactly the pointwise limits of sequences of full functions, which are particularly simple Lipschitz functions. Moreover we highlight the link between the two classical stratifications of the Borel functions by showing that the Baire class functions of some level are exactly ...
متن کاملWhen are Borel functions Baire functions ?
The following two theorems give the flavour of what will be proved. THEOREM. Let Y be a complete metric space. Then the families of first Baire class functions and of first Borel class functions from [0, 1] to Y coincide if and only if Y is connected and locally connected. THEOREM. Let Y be a separable metric space. Then the families of second Baire class functions and of second Borel class fun...
متن کاملCompact Subsets of the First Baire Class
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...
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 کاملTransfinite Sequences of Continuous and Baire 1 Functions on Separable Metric Spaces
We investigate the existence of well-ordered sequences of Baire 1 functions on separable metric spaces. Any set F of real valued functions defined on an arbitrary set X is partially ordered by the pointwise order, that is f ≤ g iff f(x) ≤ g(x) for all x ∈ X. In other words put f < g iff f(x) ≤ g(x) for all x ∈ X and f(x) 6= g(x) for at least one x ∈ X. Our aim will be to investigate the possibl...
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تاریخ انتشار 2001