scholarly journals A GAME CHARACTERIZING BAIRE CLASS 1 FUNCTIONS

2019 ◽  
Vol 85 (1) ◽  
pp. 456-466
Author(s):  
VIKTOR KISS
Keyword(s):  
Arbitrary Function ◽  
Winning Strategy ◽  
Baire Class ◽  
Polish Spaces ◽  
Class 1 ◽  
Player Game

AbstractDuparc 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.

scholarly journals Transfinite sequences of continuous and Baire class 1 functions

2002 ◽  
Vol 131 (8) ◽  
pp. 2453-2457 ◽  
Author(s):  
Márton Elekes ◽  
Kenneth Kunen
Keyword(s):  
Baire Class ◽  
Class 1

scholarly journals A Classification of Baire Class 1 Functions

1990 ◽  
Vol 318 (1) ◽  
pp. 209 ◽  
Author(s):  
A. S. Kechris ◽  
A. Louveau
Keyword(s):  
Baire Class ◽  
Class 1

Baire class 1

2001 ◽  
pp. 13-20
Author(s):  
Casper Goffman ◽  
Togo Nishiura ◽  
Daniel Waterman
Keyword(s):  
Baire Class ◽  
Class 1

scholarly journals DECOMPOSING FUNCTIONS OF BAIRE CLASS 2 ON POLISH SPACES

2020 ◽  
pp. 1-11
Author(s):  
LONGYUN DING ◽  
TAKAYUKI KIHARA ◽  
BRIAN SEMMES ◽  
JIAFEI ZHAO
Keyword(s):  
Baire Class ◽  
Polish Spaces

Analytic Ideals

1996 ◽  
Vol 2 (3) ◽  
pp. 339-348 ◽  
Author(s):  
Sławomir Solecki
Keyword(s):  
Partial Order ◽  
Polish Space ◽  
Equivalence Relations ◽  
Baire Class ◽  
Maximal Ideals ◽  
Borel Ideal ◽  
Class 1 ◽  
Long Time ◽  
Analytic Ideals ◽  
Substantial Interest

§1. Introduction. Ideals and filters of subsets of natural numbers have been studied by set theorists and topologists for a long time. There is a vast literature concerning various kinds of ultrafilters (or, dually, maximal ideals). There is also a substantial interest in nicely definable (Borel, analytic) ideals—these by old results of Sierpiński are very far from being maximal— and the structure of such ideals will concern us in this announcement. In addition to being interesting in their own right, Borel and analytic ideals occur naturally in the investigations of compact subsets of the space of all Baire class 1 functions on a Polish space (Rosenthal compacta), see [12, 18]. Also, certain objects associated with such ideals are of considerable interest and were quite extensively studied by several authors. Let us list here three examples; in all three of them I stands for an analytic or Borel ideal.1. The partial order induced by I on P(ω): X ≥I Y iff X \ Y ϵ I ([16]) and the partial order (I, ⊂)([18]).2. Boolean algebras of the form P(ω)/I and their automorphisms ([6, 5, 19, 20]).3. The equivalence relation associated with I: XEI Y iff X Δ ϵ I ([4, 14, 15,9]).In Section 4, we will have an opportunity to state some consequences of our results for equivalence relations as in 3.

scholarly journals Note on limits of simply continuous and cliquish functions

1994 ◽  
Vol 17 (3) ◽  
pp. 447-450 ◽  
Author(s):  
Janina Ewert
Keyword(s):  
Metric Space ◽  
Continuous Functions ◽  
Baire Space ◽  
Baire Property ◽  
Baire Class ◽  
Class 1 ◽  
Pointwise Limit ◽  
Separable Metric Space

The main result of this paper is that any functionfdefined on a perfect Baire space(X,T)with values in a separable metric spaceYis cliquish (has the Baire property) iff it is a uniform (pointwise) limit of sequence{fn:n≥1}of simply continuous functions. This result is obtained by a change of a topology onXand showing that a functionf:(X,T)→Yis cliquish (has the Baire property) iff it is of the Baire class 1 (class 2) with respect to the new topology.

Separate approximate continuity implies measurability

1973 ◽  
Vol 73 (3) ◽  
pp. 461-465 ◽  
Author(s):  
Roy O. Davies
Keyword(s):  
Continuum Hypothesis ◽  
Measurable Cardinal ◽  
Continuous Functions ◽  
Affirmative Answer ◽  
Baire Class ◽  
Continuous Real Function ◽  
Approximate Continuity ◽  
Class 1 ◽  
Measure Spaces ◽  
First Baire Class

It is known that a real-valued function f of two real variables which is continuous in each variable separately need not be continuous in (x, y), but must be in the first Baire class (1). Moreover if f is continuous in x for each y and merely measurable in y for each x then it must be Lebesgue-measurable (7), and this result can be extended to more general product spaces (2). However, the continuum hypothesis implies that this result fails if continuity is replaced by approximate continuity, as can be seen from the proof of Theorem 2 of (2). This makes Mišik's question (5) very natural: is a function which is separately approximately continuous in both variables necessarily Lebesgue-measurable? Our main aim is to establish an affirmative answer. It will be shown that such a function must in fact be in the second Baire class, although not necessarily in the first Baire class (unlike approximately continuous functions of one variable (3)). Finally, we show that the existence of a measurable cardinal would imply that a separately continuous real function on a product of two topological finite complete measure spaces need not be product-measurable.

scholarly journals Random-Player Maker-Breaker games

10.37236/5032 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Michael Krivelevich ◽  
Gal Kronenberg
Keyword(s):  
Perfect Matching ◽  
Winning Strategy ◽  
Hamilton Cycle ◽  
The Other ◽  
Natural Question ◽  
Vertex Connectivity ◽  
Asymptotic Values ◽  
Player Game ◽  
Matching Game ◽  
As If

In a $(1:b)$ Maker-Breaker game, one of the central questions is to find the maximal value of $b$ that allows Maker to win the game (that is, the critical bias $b^*$). Erdős conjectured that the critical bias for many Maker-Breaker games played on the edge set of $K_n$ is the same as if both players claim edges randomly. Indeed, in many Maker-Breaker games, "Erdős Paradigm" turned out to be true. Therefore, the next natural question to ask is the (typical) value of the critical bias for Maker-Breaker games where only one player claims edges randomly. A random-player Maker-Breaker game is a two-player game, played the same as an ordinary (biased) Maker-Breaker game, except that one player plays according to his best strategy and claims one element in each round, while the other plays randomly and claims $b$ (or $m$) elements. In fact, for every (ordinary) Maker-Breaker game, there are two different random-player versions; the $(1:b)$ random-Breaker game and the $(m:1)$ random-Maker game. We analyze the random-player version of several classical Maker-Breaker games such as the Hamilton cycle game, the perfect-matching game and the $k$-vertex-connectivity game (played on the edge set of $K_n$). For each of these games we find or estimate the asymptotic values of the bias (either $b$ or $m$) that allow each player to typically win the game. In fact, we provide the "smart" player with an explicit winning strategy for the corresponding value of the bias.

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