When 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 functions from [0, 1] to Y coincide if and only if for all finite sequences U1, . . . , Uq of nonempty open subsets of Y there exists a continuous function φ : [0, 1]→ Y such that φ(Ui) 6= ∅ for all i ≤ q. 0. Introduction. Given metric spacesX and Y we let Ba0(X,Y ) be the family of all continuous functions from X to Y . For all ordinals 0 < α < ω1 we define the Baire class α, denoted by Baα(X,Y ), to be the family of all limits of pointwise convergent sequences of functions from ⋃ β<α Baβ(X,Y ). A class α Borel function from X to Y (0 < α < ω1) is a function f such that f−1(G) is a Borel set of additive class α whenever G is an open subset of Y . For reference on Borel sets see [10]. We denote the family of all class α Borel functions by Boα(X,Y ). The first Baire and Borel classes do not coincide in general. The function f : [0, 1] → {0, 1} defined by f(1) = 1 and f(t) = 0 when t < 1 is of first Borel class, but clearly is not of first Baire class. The Lebesgue–Hausdorff Theorem in [10, p. 391] tells us that if X is metric and if Y is an n-dimensional cube, [0, 1], n ∈ N, or the Hilbert cube, [0, 1]N, then the first Baire and Borel classes of functions from X to Y do coincide. More general theorems of this kind has been proved. Rolewicz showed in [14] that if Y is a separable convex subset of a normal linear space, then the first Baire and Borel classes of functions from X to Y coincide. In [6] Hansell gave an extension of the Lebesgue–Hausdorff Theorem asserting that, if every continuous function from a closed subset of X to Y can be