Playing in the first Baire class

Abstract

We present a self‐contained analysis of some reduction games, which characterise various natural subclasses of the first Baire class of functions ranging from and into 0‐dimensional Polish spaces. We prove that these games are determined, without using Martin's Borel determinacy, and give precise descriptions of the winning strategies for Player I. As an application of this analysis, we get a new proof of the Baire's lemma on pointwise convergence.

A Game Characterizing Baire Class 1 Functions

“…It is clear that each y M +n was chosen according to Case (a), hence y M +n ∈ f(X α M +n ) = f(F n ), showing that the conditions of Lemma 2.1 are satisfied. The conclusion of the lemma ensures that y n → f(x), completing the analysis of Case (2). Thus, the proof of the first assertion of the theorem is complete.…”

“…By a result of Duparc, f : N N → N N is Baire class 1 if and only if Player II has a winning strategy in the eraser game. Carroy [2] proved that if f is not Baire class 1 then Player I has a winning strategy.…”

“…In this section we prove Theorem 1.1. As we mentioned before, Carroy [2] constructs a winning strategy for Player I in the eraser game (if f is not Baire class 1), and using his ideas one can construct a winning strategy for Player I in G f . We include a construction anyway to keep the article self-contained.…”

“…Note that Cases (1a) and (1b) covers all subcases of Case (1), hence it remains to show that y n → f(x) even in the following case. Case (2). n = 1 for all, but finitely many n. Let m ∈ N be large enough so that n = 1 for every n ≥ m.…”

“…Then Player I responds with 3 7 . Such an element x 1 exists using (23) and the definition of the oscillation, (2). Now let Player I play x 1 until Player II plays elements outside of the ball B(f(x 1 ), ε/7).…”