A Game Characterizing Baire Class 1 Functions

Abstract

Duparc introduced a two-player game for a function f between zero-dimensional Polish spaces in which Player II has a winning strategy iff f is of Baire class 1. We generalize this result by defining a game for an arbitrary function f : X → Y between arbitrary Polish spaces such that Player II has a winning strategy in this game iff f is of Baire class 1. Using the strategy of Player II, we reprove a result concerning first return recoverable functions.

Games characterizing limsup functions and Baire class 1 functions

“…It is then straightforward to check that the strategy of Player II defined this way is winning. ✷ Unlike the eraser game (see Kiss [7]), as we will show below, the game Γ( f ) need not be determined.…”

“…✷ Remark 4. 3 The above construction of the winning strategy for Player I is similar to that in Kiss [7]. In both cases Player I zooms in on a particular element of C until Player II triggers a switch to another element.…”

“…The so-called eraser game characterizing Baire class 1 functions from the Baire space to itself was constructed in Duparc [4]. Carroy [2] showed that the eraser game is determined, and Kiss [7] generalized the characterization to functions of arbitrary Polish spaces. Game characterizations of several other classes of functions have been considered in Semmes [12], Carroy [2], and Nobrega [11].…”

Games characterizing certain families of functions

“…Recently, Kiss [13] introduced a game characterizing Baire class 1 functions between arbitrary two Polish spaces. This improved the results by Duparc [7] and Carroy [6] that have been mentioned above.…”