Proceedings of the Ninth Prague Topological Symposium Contributed papers from the symposium held in Prague, Czech Republic, August 19–25, 2001

TRANSFINITE SEQUENCES OF CONTINUOUS AND BAIRE 1 FUNCTIONS ON SEPARABLE METRIC SPACES

MÁRTON ELEKES

Abstract. 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) ≠ 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 < ω_{1} (see [2, §24. III.2’]). Moreover, P. Komjáth proved in [1] that the corresponding question concerning Baire α functions for 2 ≤ α < ω_{1} is independent of ZF C.

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, ϱ) 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 ω_{1} of Baire 1 functions.

2000 Mathematics Subject Classification. 26A21.

Key words and phrases. Baire 1 function, well-ordered sequence, metric spaces.

This is a research announcement. A complete article, written jointly with Kenneth Kunen, will be published elsewhere.

Márton Elekes, Transfinite sequences of continuous and Baire 1 functions on separable metric spaces, Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001), pp. 91–92, Topology Atlas, Toronto, 2002; arXiv:math.GN/0204124.

Theorem. The following statement: ‘There exists a separable metric space on which there exists a well-ordered sequence of length ω_{2} of Baire 1 functions’ is independent of ZF C + ¬CH.

Remark. During and after the conference Kenneth Kunen answered one of my questions, and also improved some of the results and proofs. These results will appear in a forthcoming joint paper.

References

[1] Péter Komjáth, Ordered families of Baire-2-functions, Real Anal. Exchange 15 (1989/90), no. 2, 442–444. MR 91d:26007

[2] K. Kuratowski, Topology. Vol. I, Academic Press, New York, 1966. MR 36 #840

Department of Analysis, Eötvös Loránd University, Budapest, Pázmány Péter sétány 1/c, 1117, Hungary

   E-mail address: emarci@cs.elte.hu