Point-set games and functions with the hereditary small oscillation property

Given a metric space $X$, we consider certain families of functions $f:X\to\mathbb{R}$ having the hereditary oscillation property HSOP and the hereditary continuous restriction property HCRP on large sets. When $X$ is Polish, among them there are families of Baire measurable functions, $\overline{\mu}$-measurable functions (for a finite nonatomic Borel measure $\mu$ on $X$) and Marczewski measurable functions. We obtain their characterizations using a class of equivalent point-set games. In similar aspects, we study cliquish functions, SZ-functions and countably continuous functions.


Introduction
In topology several kinds of games between two players have various important applications.They can characterize certain classes of sets or functions in a transparent way.For some survey articles, see [1,4,8].In our paper we present characterizations of several natural families of real-valued functions defined on a metric space X, via point-set games.We continue ideas initiated in [2] where point-set games were used to characterize Baire measurable functions on Polish spaces and Lebesgue measurable functions on R k .The motivation of [2] came from the article of Kiss [10] who obtained a new game characterization of Baire 1 functions.Here our approach is more general.We distinguish classes of functions having the hereditary continuous restriction property HCRP and the hereditary small oscillation property HSOP with respect to various kinds of large sets.Two types of point-set games are useful: one of them is G 1 (Σ, f ) that was considered in [2] in particular cases, and the other game G <ω (Σ, f ) is a slight modification of G 1 (Σ, f ).These are examples of the so-called selection games, which are well-represented in the literature; see e.g.[13].In particular, a general versions of G 1 (Σ, f ) and G <ω (Σ, f ) are considered by Clontz in the recent paper [6].We show that these two games are equivalent and, in several settings, determined.However, we leave an open question whether they are determined in general.
The paper is organized as follows.Section 2 is devoted to the definitions of games G 1 (Σ, f ) and G <ω (Σ, f ) and their relationships.In particular, we show that these games are equivalent.In Section 3 we introduce properties HCSP and HSOP and we prove the basic Theorem 3.1 which describes connections between HCRP and HSOP (for f : X → R with respect to a family Σ) and the existence of winning strategies of Players I and II in the games G 1 (Σ, f ) (so also in G <ω (Σ, f )).In Section 4 we show applications of our results to some kinds of measurable functions having the HSOP with respect to the corresponding families Σ of large sets.Other interrelations with the topic are discussed in Sections 5 and 6.This part of our paper is concerned with cliquish functions, SZ-functions and countably continuous functions.

A scheme of games
Let X be a metric space.Assume that Σ is a non-empty family of non-empty subsets of X and f : X → R is fixed.We will define two point-set games G <ω (Σ, f ) and G 1 (Σ, f ).Note that G 1 (Σ, f ) was studied in [2].
The game G <ω (Σ, f ) is defined as follows.At the initial step of G <ω (Σ, f ), Player I plays P ∈ Σ, then Player II plays a set P 0 .At the nth step, n ≥ 1, Player I plays a finite sequence x i : k n−1 < i ≤ k n (where k 0 = 0 and k n is a strictly increasing sequence of naturals), and Player II plays a P n : with the rules that for each integer n ≥ 0: Otherwise, Player I wins.
The game G 1 (Σ, f ) is similar to G <ω (Σ, f ) with one difference: for every n > 0 we assume that k n = n, i.e. we have Note that the finite sequence played by Player I in G <ω is non-empty.This distinguishes this game from the game G <ω defined in [6].
Remark 2.2.If Player I has a winning strategy in the game G 1 (Σ, f ) then he has also a winning strategy in the game G <ω (Σ, f ).If Player II has a winning strategy in the game G <ω (Σ, f ), then he has also a winning strategy in the game G 1 (Σ, f ).
Proposition 2.3.Assume that Σ is a non-empty family of non-empty subsets of a metric space (X, d) and f ∈ R X .If Player I has a winning strategy in the game G <ω (Σ, f ), then he has also a winning strategy in the game G 1 (Σ, f ).
Proof.Let $ <ω be a winning strategy of Player I in the game G <ω (Σ, f ).We will define a strategy $ 1 for Player I in the game G 1 (Σ, f ) and we will show that this strategy is winning.
To show that $ 1 is a winning strategy, suppose to the contrary that there exists a sequence P n n∈N of sets from Σ such that, if x n := $ 1 (P 1 , P 2 , . . ., P n ) for n ∈ N, then (1) lim n→∞ x n = x and lim Since $ <ω is a winning strategy for Player I in G <ω (Σ, f ), we can fix ε > 0 such that, if x ′ k n−1 +1 , . . ., x ′ kn := $ <ω (P 1 , P 2 , . . ., P n ), then for each N ∈ N there exist n > N and j ∈ {k By (1) we can fix N 0 ∈ N such that (3) Then, by (2) we can find m > N 0 and j ∈ {k m−1 + 1, . . ., k m } such that (4) d(x ′ j , x) > ε.Without loss of generality we can assume that (5) x since other cases are analogous.Then, by the formula defining $ 1 we obtain ) for all j ∈ {k m−1 + 1, . . ., k m }, and consequently, by ( 6) which is a desired contradiction.
Proposition 2.4.Assume that Σ is a non-empty family of non-empty subsets of a metric space (X, d) and f ∈ R X .If Player II has a winning strategy in the game G 1 (Σ, f ), then he has also a winning strategy in the game G <ω (Σ, f ).
We will verify that $ <ω is a winning strategy for Player II in the game G <ω (Σ, f ).Suppose to the contrary that there exists a sequence P, x 1 , x 2 , . . ., x kn−1 , x kn . . .such that Player I wins the game G <ω (Σ, f ) in which he plays x k n−1 +1 , . . ., x kn in the nth move, and Player II responds with P n := $ <ω (P, x 1 , . . ., x kn ).Since $ 1 is a winning strategy for Player II in the game G 1 (Σ, f ), there exists x ∈ X such that lim n xn = x and lim n f (x n ) = f (x).But Player I wins the game G <ω (Σ, f ) with the moves x k n−1 +1 , . . ., x kn for n ∈ N, so either the sequence x n does not converge to x, or f (x n ) does not converge to f (x).Therefore, we can fix ε > 0 such that for each N ∈ N there exists n > N such that d(x n , x) > ε.
We say that two games 1 and 2 are equivalent whenever each of players has a winning strategy in the game 1 if and only if he has a winning strategy in the game 2. Thus Propositions 2.3 and 2.4 yield the following corollary.
Corollary 2.5.Assume that Σ is a non-empty family of non-empty subsets of a metric space (X, d) and f ∈ R X .Then the games G 1 (Σ, f ) and G <ω (Σ, f ) are equivalent.
We can also consider the following modifications of the games G 1 (Σ, f ) and G <ω (σ, f ).Let λ be an increasing sequence of natural numbers.The definition of the game G λ (Σ, f ) is the same as the definition of G <ω (Σ, f ) with the condition: at the nth step of the game G m (Σ, f ), where n ≥ 1, Player I plays a finite sequence x i : m(n − 1) < i ≤ mn and Player II plays a set P n as follows: It is easy to observe that: • If Player I has a winning strategy in the game G 1 (Σ, f ) then he has also a winning strategy in the game G λ (Σ, f ), for any increasing sequence λ.• If Player II has a winning strategy in the game G <ω (Σ, f ), then he has also a winning strategy in the game G λ (Σ, f ) for every λ.Therefore, all games G λ (Σ, f ) are equivalent to the game G 1 (Σ, f ) (so also to the game G <ω (Σ, f )).
In the next sections we give several examples where the games G 1 (Σ, f ) (so, also G <ω (Σ, f )) are determined for functions f from special classes of functions with the respectively chosen Σ.The following question seems to be interesting in this context.Problem 2.6.Do there exist a family Σ of non-empty subsets of R and a function f : R → R for which the game G 1 (Σ, f ) is not determined?

Functions with the hereditary small oscillation property on large sets
Let g : A → R with A ⊆ X. Recall the notion of the oscillation osc(g, x) of g at a point x ∈ A. Namely, let osc(g, x) := lim ε→0 + diam(g(A ∩ B(x, ε)) where B(x, ε) denotes the open ball centered at x with radius ε.We know that g is continuous at x ∈ A if and only if osc(g, x) = 0.
The collection Σ will play a role of large subsets of X.We say that a family Σ is dense if, for each P ∈ Σ and every ball B(x, r) with x ∈ P there exists Q ∈ Σ contained in P ∩ B(x, r).
Given a family of non-empty subsets of X, we say that a function f : X → R has: • the hereditary continuous restriction property (HCRP) with respect to Σ whenever for each P ∈ Σ there exists a subset Q ⊆ P in Σ such that f ↾ Q is continuous; • the hereditary small oscillation property (HSOP) with respect to Σ whenever for every α > 0 and each P ∈ Σ there exists a subset Q ⊆ P in Σ such that osc(f ↾ Q, x) < α for all x ∈ Q.We say that a family F ⊆ R X has the HCRP (respectively, the HSOP) with respect to Σ whenever every function f in F has a property with the same name.We use this terminology to simplify the notation in our results.Note that the HCRP was used in [3] in an implicit way without this special notation.It is a stronger version of the continuous restrictions property (CRP) introduced by Rec law in [12].
Clearly, the hereditary continuous restriction property for f implies its hereditary small oscillation property.In the next section, we will discuss some examples for which the reverse implication holds.Now, our aim is to prove the following general result.We will consider the following statements: (i) f has the HCRP with respect to Σ; (ii) f has the HSOP with respect to Σ; (iii) Player II has a winning strategy in the game G 1 (Σ, f ); (iv) Player I has a winning strategy in the game G 1 (Σ, f ).
Theorem 3.1.Assume that (X, d) is a complete metric space and Σ is a dense family whose members are non-empty closed subsets of X.Then (ii) ⇔ (iii) and ¬(ii) ⇔ (iv).This means that the game Assume that f has the HSOP with respect to Σ. Let Player I choose P ∈ Σ.Let Player II pick a subset P 0 ∈ Σ of P such that osc(f ↾ P 0 , x) < 1 for x ∈ P 0 .Assume that Player I has made his kth move.So, points x 1 , x 2 , . . ., x k have been chosen.Since the family Σ is dense, there is Q ∈ Σ such that By the assumption on f , there is k for all x, x ′ ∈ P k .Let Player II play P k at his kth move.Then Player I will choose x k+1 ∈ P k and so on.
Observe that x i is a Cauchy sequence.Indeed, fix positive integers N and k > N. Then The remaining details are left to the reader.
Since X is complete, there exists x ∈ X such that lim i x i = x.Since the sets P k are closed, k for i > k and consequently, lim i f (x i ) = f (x)."¬(ii) ⇒ (iv)".Suppose that f has not the HSOP with respect to Σ. So, there exist a set P ∈ Σ and α > 0 such that, for each Q ⊆ P in Σ, there exists x ∈ Q with osc(f ↾ Q, x) ≥ α.Let Player I play as follows.Firstly, he chooses P as above.Next Player II picks P 0 ∈ Σ, P 0 ⊆ P , and Player I chooses any x 1 ∈ P 0 .Assume that for some integer k > 0 a point x k−1 and a set P k have been chosen.Then Player I picks any x ∈ P k and he considers two cases.
and Player II chooses Hence the sequence f (x n ) is not convergent and Player I wins the game G 1 (Σ, f ).Thus Player I has the winning strategy in this game.
Finally, observe that in both cases, (ii) or ¬(ii), one of the players has a winninig strategy in the game G 1 (Σ, f ), so the game is determined.Remark 3.2.Notice that the assumption that any set in Σ is closed and X is complete is only used in the proof of the implication (ii) ⇒ (iii).
Since the implication (i) ⇒ (ii) holds for any f : X → R, we have the following fact.
Corollary 3.3.Assume that (X, d) is a complete metric space, Σ is a dense family whose members are non-empty closed subsets of X.If the implication (ii) ⇒ (i) holds for a function f : X → R, then the statements (i),(ii),(iii),¬(iv) are equivalent.

Applications
Here we discuss some examples of families of functions with the hereditary continuous restriction property on large sets.Three of them come from the article [3] where the authors considered characterizations by this property for certain functions that are measurable with respect to some known σ-algebras of sets.Now, let X be a Polish space without isolated points.By Perf we denote the family of all nonempty perfect subsets of X.Consider three families of measurable functions.
Baire measurable functions.These are functions f : X → R which are measurable with respect to σ-algebra of sets with the Baire property.It is well known that if X is complete then Baire measurable functions are exactly those functions f whose restrictions f ↾ G to a residual (comeager) set G are continuous (see e.g.[9,Thm 8.38]).It was proved in [3,Thm 9] that f is Baire measurable if and only if it has the HCRP with respect to G δ sets G ⊆ X having the property: there is a nonempty open set U such that G is residual in U.The family of these G δ sets will be denoted by G Res .
Functions measurable with respect to a measure.Let M be the collection of sets which are measurable with respect to the completion µ of a finite nonatomic Borel measure µ on X. and let Perf + stand for the family of all perfect subsets of X with positive measure.It was proved in [3,Thm 8] that f : X → R is M-measurable if and only if it has the HCRP with respect to Perf + .Marczewski measurable functions.These are functions f : X → R which are measurable with respect to σ-algebra of (s)-sets E ⊆ X with the property: for each P ∈ Perf there exists Q ∈ Perf such that either Q ⊆ P ∩ E or Q ⊆ P \ E, cf.[14] and [3].In [14] the following nice characterization was established (see also [3]): a function f is (s)-measurable if and only if it has the HCRP with respect to Perf.HSOP versus HCRP.We are going to prove that for the above kinds of measurable functions, the HSOP is equivalent to the respective HCRP.Proposition 4.1.Let X be a Polish space without isolated points and f : X → R. Then: (a) if f has the HSOP with respect to Perf, then it is Marczewski measurable; (b) given a finite nonatomic Borel measure µ on X, if f has the HSOP with respect to Perf + , then it is measurable with respect to µ; (c) if f has the HSOP with respect to G Res , then it is Baire measurable.
Consequently, in all these cases, the HSOP is equivalent to the respective HCRP for f .Proof.In fact, we will use the above-mentioned characterizations of the three kinds of measurability.So, we will show that the HSOP implies the respective HCRP, as it is stated in the final assertion.
Ad (a).Let P ∈ Perf.Using the assumption, pick a perfect subset Q of P such that osc(f ↾ Q, x) < 1 for each x ∈ Q.Then find two disjoint perfect subsets Q 0 , Q 1 of Q, of diameter less than 1.Assume that for n ∈ N, perfect pairwise disjoint sets Q s , s ∈ {0, 1} n , have been chosen where each of them has diameter less than 1/n, and osc(f ↾ Q s , x) < 1/n for every x ∈ Q s .For any fixed Q s find two disjoint perfect subsets Q s⌢0 and Q s⌢1 of Q s , of diameter less than 1/(n + 1), with osc(f ↾ Q s⌢i , x) < 1/(n + 1) for all x ∈ Q s⌢i and i ∈ {0, 1}.Finally, define Note that Q ⋆ is a perfect subset of P and f ↾ Q ⋆ is continuous since its oscillation at every point of Q ⋆ is zero.
Ad (b).We should be more careful in comparison with the previous case.Fix a perfect set P of positive measure.In the first step, using the assumption, we consider a maximal disjoint family J 1 of perfect sets Q ⊆ P of positive measure such that osc(f ↾ Q, x) < 1 for all Q ∈ J 1 and x ∈ Q.Note that J 1 is countable and µ( J 1 ) = µ(P ).Pick a finite family J * 1 ⊆ J 1 such that for S 1 := J * 1 we have µ(P \ S 1 ) < µ(P )/2 2 .Then osc(f ↾ S 1 , x) < 1 for each x ∈ S 1 .In the next step, in a similar way, we select finite disjoint family of perfect subsets of S 1 whose union, called S 2 , satisfies the conditions µ(S 1 \ S 2 ) < µ(P )/2 3 and osc(f ↾ S 2 , x) < 1/2 for each x ∈ S 2 .In such a way we obtain a decreasing sequence of perfect sets S n such that for every n ∈ N we have µ(S n \ S n+1 ) < µ(P )/2 n+2 and osc(f ↾ S n , x) < 1 n for each x ∈ S n .
Then note that the set S := n∈N S n is closed and µ(S) > µ(P )/2.Also, osc(f ↾ S, x) = 0 for each x ∈ S. Finally, we can select a perfect set Q ⊆ S of positive measure.Then f ↾ Q is continuous.Ad (c).Fix a G δ set G that is contained in a nonempty open set U where M := U \ G is meager (that is, of the first category) of type F σ .Using the assumption, we can find, for each k ∈ N, a maximal disjoint family J k of open subsets of U such that for every V ∈ J k there is a meager set Next, we define an increasing sequence of meager sets T n (for n ∈ N) of type F σ , contained in U, such that osc(f ↾ (U \ T n ), x) < 1/n for each n ∈ U \ T k .Namely, let T 1 := M ∪ W 1 , and T n+1 := T n ∪ i≤n W i .Finally, observe that T := n∈N T n is a meager subset of U, of type F σ and H := U \ T is a set, residual in U, of type G δ , contained in G, with osc(f ↾ H, x) = 0 for each x ∈ H, as desired.
Remark 4.2.Statements (b) and (c) in the above proposition can also be deduced from [2, Lemma 10].However, for the reader's convenience, we have decided to include here the direct proofs of these facts.
Observe that, if X is a Polish space without isolated points, the families G Res , Perf + and Perf are dense.Additionally, Perf + and Perf consist of closed sets.So, by Proposition 4.1 and Theorem 3.1, we can infer the following corollaries.
Corollary 4.3.Let X be a Polish space without isolated points.Let f : X → R. Given a finite nonatomic Borel measure µ on X, let Σ := Perf + .Then the game G 1 (Σ, f ) is determined and • if f is measurable with respect to the completion of µ then Player II has a winning strategy in the game G 1 (Σ, f ), • if f s not measurable with respect to the completion of µ then Player I has a winning strategy in the game G 1 (Σ, f ).
Note that the the same result was established in [2] in a bit different manner.We have a small problem with Baire measurable functions since the respective family G Res consists of G δ sets, and the closedness of sets was used in the proof of Theorem 3.1 (cf.Remark 3.2).We will solve this problem by the following modification.
Corollary 4.5.Let X be a Polish space without isolated points.Let f : X → R and Σ := G Res .Then the game G 1 (Σ, f ) is determined and • if f is Baire measurable then Player II has a winning strategy in the game G 1 (Σ, f ), • if f is not Baire measurable then Player I has a winning strategy in the game G 1 (Σ, f ).
Proof.By Remark 3.2, only the first implication has to be proved.Assume that f : X → R has the Baire property.Then there is a meager Then Player I will choose x k+1 ∈ P k and so on.As in the proof of Theorem 3.1, one can verify that the sequence x i converges to some x ∈ X.Note that x i ∈ cl(P k ) for every i > k, hence x ∈ i∈N cl(P i ) and i∈N cl(P i ) ∩ M = ∅, so x ∈ X \ M. Since all points x i belong to P 0 ⊆ X \ M and f ↾ (X \ M) is continuous at x, so lim i f (x i ) = f (x).Note that in [2] the Baire measurability of a function was characterized in the language of the winning strategy of Player II in the game G 1 (Σ, f ) where Σ consists of Baire nonmeager sets.

Cliquish functions
Let X be a topological space.Recall that a function f : X → R is: • cliquish if for each non-empty open set W and ε > 0 there is a non-empty open set U ⊆ W with diam(f (U)) < ε; see [15]; • pointwise discontinuous if the set C(f ) of continuity points of f is dense in X; see [11].It is well-known and easy to show that each pointwise discontinuous function is also cliquish.Moreover, if X is a Baire space (i.e., X is non-meager in itself) then those notions are equivalent.The following fact results directly from the definition.
The second part of the assertion follows from the fact that if X is separable then there exists a SZ-function f ∈ R X , cf. [5], and any such an f has the HSOP but it has not the HCRP with respect to Σ. Corollary 6.3.Assume that c is a regular cardinal, X is an uncountable Polish space and Σ = [X] c .Then for any f ∈ R X , Player II has a winning strategy in the game G 1 (Σ, f ).
Proof.At the beginning of the play, Player I chooses a set P ∈ [X] c .For every n ∈ N let A n denote a maximal family of pairwise disjoint subsets A ⊂ P such that |A ∩ B(x, ε)| = c for every ε > 0 and x ∈ A (i.e., A is c-dense-in-itself) and |f (x) − f (x ′ )| < 1 n for x, x ′ ∈ A. Observe that B n := P \ A n is of size less than c.Let B := n B n .Since c is a regular cardinal, A := P \ B has cardinality c, so there exists x 0 ∈ A such that |A ∩ B(x 0 , ε)| = c for every ε > 0. For every k ∈ N there exists A k ∈ A k with x 0 ∈ A k .Now, Player II chooses in his kth move the set P k := A k ∩ B(x 0 , 1 k ).Then, regardless of the choice of points x k ∈ P k−1 by Player I, we have lim n x n = x 0 and |f (x) − f (x 0 )| < 1 k , hence lim n f (x n ) = f (x 0 ).To conclude, the above is a winning strategy for Player II.
Corollary 4.4.Let X be a Polish space without isolated points.Let f : X → R and Σ := Perf.Then the game G 1 (Σ, f ) is determined and• if f is Marczewski measurable then Player II has a winning strategy in the game G 1 (Σ, f ), • if f is not Marczewski measurable then Player I has a winning strategy in the game G 1 (Σ, f ).