Transfinite 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 possible length of the increasing or decreasing well-ordered sequences of functions in F with respect to this order. A classical theorem of Kuratowski asserts, that if F is the set of continuous or Baire 1 functions defined on a Polish space X, then there exists a monotone sequence of length ξ in F iff ξ < ω1 (see [2, §24. III.2’]). Moreover, P. Komjáth proved in [1] that the corresponding question concerning Baire α functions for 2 ≤ α < ω1 is independent of ZFC. In the present paper we investigate what happens if we drop the condition of completeness and replace the Polish space X by a separable metric space. Our main results are the following. Let d(X) denote the density of a space X. Theorem. Let (X, ̺) be a metric space. Then there exists a well-ordered sequence of length ξ of continuous real-valued functions defined on X iff ξ < d(X). Corollary. A metric space is separable iff every well-ordered sequence of continuous functions defined on it is countable. Theorem. There exists a separable metric space on which there exists a well-ordered sequence of length ω1 of Baire 1 functions. 2000 Mathematics Subject Classification. 26A21.