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Journal of Applied Analysis Vol. 11, No. 2 (2005), pp. 153–
ON ADDITIVE ALMOST CONTINUOUS
FUNCTIONS UNDER CPA
game prism
K. CIESIELSKI and J. PAWLIKOWSKI
Received August 25, 2003 and, in revised form, February 20, 2004
Abstract. We prove that the Covering Property Axiom CPAgameprism, which holds in the iterated perfect set model, implies that there ex- ists an additive discontinuous almost continuous function f : R → R whose graph is of measure zero. We also show that, under CPAgameprism, there exists a Hamel basis H for which, E+(H), the set of all linear combinations of elements from H with positive rational coefficients, is of measure zero. The existence of both of these examples follows from Martin’s axiom, while it is unknown whether either of them can be constructed in ZFC. As a tool for the constructions we will show that CPAgameprism implies its seemingly stronger version, in which ω 1 -many games are played si- multaneously.
2000 Mathematics Subject Classification. Primary 26A15, 26A30; Secondary 03E35. Key words and phrases. Additive, almost continuous, Hamel basis, Covering Property Axiom, CPA. The work of the first author was partially supported by 2002/03 West Virginia University Senate Research Grant. The second author wishes to thank West Virginia University for its hospitality in years 1998–2001, where the results presented here were obtained.
ISSN 1425-6908 ©c Heldermann Verlag.
154 K. CIESIELSKI AND J. PAWLIKOWSKI
- Preliminaries and axiom CPAgameprism
Our set theoretic terminology is standard and follows that of [1]. In particular, |X| stands for the cardinality of a set X and c = |R|. The Cantor set 2ω^ will be denoted by a symbol C. We use term Polish space for a complete separable metric space without isolated points. For a Polish space X, symbol Perf(X) will stand for a collection of all subsets of X homeomorphic to the Cantor set C. For a fixed 0 < α < ω 1 and 0 < β ≤ α a symbol πβ will stand for the projection from Cα^ onto Cβ^. In what follows we will consider R as a linear space over Q. For Z ⊂ R its linear span with respect to this structure will be denoted by LIN(Z). A subset H of R is a Hamel basis provided it is a linear basis of R over Q, that is, it is linearly independent and LIN(H) = R. Axiom CPAgameprism was introduced by the authors in [5], where it is shown that it holds in the iterated perfect set model. Also, CPAgameprism is a simpler version of the axiom CPA which is described in a monograph [9]. For the reader’s convenience, we will restate CPAgameprism in the next few paragraphs. For 0 < α < ω 1 let Φprism(α) be the family of all continuous injections f : Cα^ → Cα^ with the property that
f (x) β = f (y) β ⇔ x β = y β for all β ∈ α and x, y ∈ Cα.
Functions Φprism(α) are called projection-keeping homeomorphisms. (Com- pare [11].) Let Pα = {range(f ) : f ∈ Φprism(α)} and Pω 1 =
0 <α<ω 1 Pα. We will refer to elements of Pω 1 as iterated perfect sets. (In [17] the elements of Pα are called I-perfect, where I is the ideal of countable sets.) The simplest elements of Pα are perfect cubes, that is, the sets of the form C =
β<α Cβ^ , where Cβ ∈ Perf(C) for each β < α.
Claim 1.1. Let 0 < α < ω 1. If G is a Borel second category subset of Cα then G contains a perfect cube. In particular, if G is a Borel countable cover of Cα^ then there is a G ∈ G which contains an E ∈ Pα.
An argument for the claim can be found in [4, Claim 3.2] or [9, Claim 1.1.5]. The only properties of the iterated perfect sets that we will use in this paper are listed in the next three lemmas.
Lemma 1.2. For every E ∈ Pω 1 , a Polish space X, and a continuous func- tion f : E → X there exists a P ∈ Pω 1 such that P ⊂ E and f [P ] is either a singleton or it is homeomorphic to the Cantor set.
Lemma 1.2 follows immediately from [9, Lemma 3.2.5] (see also [6, Lemma 1.1] or [7, Lemma 2.4]) which is a particular case of [11, Theo- rem 20].
156 K. CIESIELSKI AND J. PAWLIKOWSKI
P ∈ Perf(X) we mean that Q = f [E], where f is a witness function for P and E ⊂ dom(f ) is an iterated perfect set. Using the fact that Φprism(α) is closed under the composition, it is easy to see that we can always assume that a witness function of a prism is always defined on the entire space Cα for an appropriate α. Let Perf∗(X) stand for the family of all sets P such that either P ∈ Perf(X) or P is a singleton in X. In what follows we will consider singletons as constant prisms, that is, with the constant coordinate function from Cα onto the singleton. In particular, a subprism of a constant prism is the same singleton. Consider the following game GAMEprism(X) of length ω 1. The game has two players, Player I and Player II. At each stage ξ < ω 1 of the game Player I can play an arbitrary prism Pξ ∈ Perf∗(X) and Player II must respond with a subprism Qξ of Pξ. The game 〈〈Pξ , Qξ〉 : ξ < ω 1 〉 is won by Player I provided
ξ<ω 1 Qξ^ =^ X; otherwise the game is won by Player II. A strategy for Player II is any function S such that S(〈〈Pη, Qη〉 : η < ξ〉, Pξ) is a subprism of Pξ, where 〈〈Pη, Qη〉 : η < ξ〉 is any partial game. (We abuse here slightly the notation, since function S depends also on the implicitly given coordinate functions fη making each Pη a prism.) A game 〈〈Pξ , Qξ〉 : ξ < ω 1 〉 is played according to a strategy S for Player II when Qξ = S(〈〈Pη, Qη〉 : η < ξ〉, Pξ) for every ξ < ω 1. A strategy S for Player II is a winning strategy for Player II provided Player II wins any game played according to the strategy S. Here is the axiom.
CPAgameprism: c = ω 2 and for any Polish space X Player II has no winning strategy in the game GAMEprism(X).
In what follows we will use the following prism density fact, which proof can be found in [8, Lemma 2.1] or in [9, Lemma 5.1.5].
Lemma 1.5. Let M ⊂ R be a sigma-compact and linearly independent. Then for every prism P in R there exist a subprism Q of P and a compact subset R of P \ M such that M ∪ R is a maximal linearly independent subset of M ∪ Q.
We will also use the following fact.
Fact 1.6. CPAgameprism implies that cof(M) = ω 1 , where cof(M) is the cofi- nality of the ideal of meager sets.
Proof. It is proved in [4, Corollary 4.3] (see also [9, Corollary 1.1.3]) that CPAcube implies that cof(N ), the cofinality of the ideal of measure zero sets, is equal to ω 1 , while it is well known that cof(N ) = ω 1 implies that
ON ADDITIVE ALMOST CONTINUOUS FUNCTIONS UNDER CPAgameprism 157
cof(M) = ω 1. To finish the argument, it is enough to recall that CPAgameprism implies CPAcube. (See e.g. [5] or [9].)
- Multi-games
For a non-empty collection X of pairwise disjoint Polish spaces consider the following “simultaneous” two-player game GAMEprism(X ) of length ω 1. At each stage ξ < ω 1 of the game Player I can play a prism Pξ ∈ Perf∗(X) from an arbitrarily chosen X ∈ X. Player II responds with a subprism Qξ of Pξ. The game 〈〈Pξ, Qξ〉 : ξ < ω 1 〉 is won by Player I provided ⋃
ξ<ω 1
Qξ =
X ;
otherwise the game is won by Player II. Thus, for any Polish space X the games GAMEprism(X) and GAMEprism({X}) are identical.
Theorem 2.1. Let X of size ≤ ω 1 be a non-empty collection of pairwise disjoint Polish spaces. Then CPAgameprism is equivalent to
CPAgameprism(X ): Player II has no winning strategy in GAMEprism(X ).
Proof. We will leave the implication “CPAgameprism(X ) implies CPAgameprism” with- out a proof, since it will not be used in the sequel. Its proof can be found in [9]. To see the converse implication assume that CPAgameprism holds and let I =
[0, 1]. Let L = {xξ : ξ < ω 1 } be a Luzin set in I, that is, such that |L∩N | ≤ ω for every nowhere dense subset N of I. The existence of such a set under CPAgameprism follows from Fact 1.6. Let κ = |X | ≤ ω 1 and let {Xη : η < κ} be an enumeration of X. We will identify each Xη, η < κ, with a Gδ subset of {xη} × Iω^ homeomorphic to it. Now, let S 0 be a Player II strategy in the game GAMEprism(X ). We will modify it to a Player II strategy S in the game GAMEprism(I × Iω) in the following way. First, for every prism P in I × Iω^ let R(P ) be its subprism such that
either R(P ) ⊂ Xη for some η < κ or R(P ) ∩
X = ∅.
To choose such R(P ) first choose subprism R of P such that its first coordi- nate projection π[R] is nowhere dense in I. (This can be done, for example, applying Lemma 1.2.) So, π[R] contains at most countably many points xη. Thus, by Claim 1.1, there is a subprism R 1 of R such that either π[R 1 ] is disjoint with L or there is an η < κ such that π[R 1 ] = {xη}. In the first case we put R(P ) = R 1. In the second case we use Claim 1.1 to find a subprism R(P ) of R 1 such that either R(P ) ⊂ Xη or R(P ) ∩ Xη = ∅.
ON ADDITIVE ALMOST CONTINUOUS FUNCTIONS UNDER CPAgameprism 159
Theorem 3.1. CPAgameprism implies that for every dense Gδ set G ⊂ R such that 0 ∈ G there exists an additive discontinuous almost continuous function f : R → R whose graph is a subset of (R × G) ∪ (G × R) = (Gc^ × Gc)c.
Using Theorem 3.1 with G of measure zero we obtain immediately the following corollary.
Corollary 3.2. CPAgameprism implies that there exists a discontinuous, almost continuous, additive function f : R → R whose graph is of measure zero.
Notice that if Lm, for 0 < m < ω, is the collection of all functions ` : Rm^ → R given by a formula
`(x 0 ,... , xm− 1 ) =
i 160 K. CIESIELSKI AND J. PAWLIKOWSKI
(b) If P is compact and G is open then G[P ] is open. (c) If P =
i<ω Pi^ and^ G^ =^
n<ω Gn^ then^ G[P^ ] =^
i,n<ω Gn[Pi]. (d) If P is sigma compact and G is a Gδ set then G[P ] is also a Gδ set. (e) If G[Pn] is a dense Gδ set for every n < ω then so is G[
n<ω Pn]. (f) G[P ][S] = G[P + S].
Proof. (a) follows immediately from the second part of (3.3) while (b) from its first part. To see (c) notice that, by (3.3),
G[P ] =
i<ω
{x ∈ R : x−Pi ⊂ G} =
i,n<ω
{x ∈ R : x−Pi ⊂ Gn} =
i,n<ω
Gn[Pi].
So, (d) follows immediately from (b), while (e) is an easy consequence of (c). Note also that
G[P ][S]c^ = G[P ]c^ + S = Gc^ + P + S = G[P + S]c
so (f) holds.
Recall that for a Polish space X the space C(X) of continuous functions from X into R is considered with the metric of uniform convergence.
Lemma 3.4. Let X be a Polish space and ¯x ∈ K¯ ∈ Perf(X). For every dense Gδ-set G ⊂ R and a prism P in C(X) there exist a subprism Q of P and a K ∈ Perf( K¯) with x¯ ∈ K such that G[LIN(RK (Q))] is a dense Gδ subset of R, where RK (Q) = {h(x) : h ∈ Q & x ∈ K}.
Proof. Let U be a countable family of open subsets of R with the property that G =
U and fix a countable basis B for R. For 0 < m < ω let Lm be the set of all functions ` defined as in (3.1) and put L =
0 162 K. CIESIELSKI AND J. PAWLIKOWSKI
s ∈ 2 Ak+1^ and r ∈ 2 k+1^ such that
|h(g)(x) − h(gs)(xr)| ≤ |h(g)(x) − h(gs)(x)| + |h(gs)(x) − h(gs)(xr)| < 2 δ,
while h(gs)(xr) ∈ Y. So, bk+1 − k+1[Rk+1] ⊂ bk+1 − B(Z, ε) ⊂ Uk+1. Thus, bk+1 ∈ Uk+1(k+1[Rk+1]) ∩ Bk+1.
As a corollary, needed in the proof but also interesting on its own, we conclude the following.
Lemma 3.5. For every dense Gδ subset G of R and for every prism P in R there exists a subprism Q of P such that G[LIN(Q)] is a dense Gδ subset of R.
Proof. Let f ∈ Φprism(α) be such that P = f [Cα] and let h : R → C(R) be given by h(r)(x) = r + x. Then h[P ] is a prism in C(R) witnessed by h ◦ f. By Lemma 3.4 there exist a subprism Q 0 = h ◦ f [E] of h[P ] and a K ∈ Perf(R) with 0 ∈ K such that Z = G[LIN({g(x) : g ∈ Q 0 & x ∈ K})] is dense in R. But then Q = f [E] = h−^1 (Q) is a subprism of P and, since 0 ∈ K,
Z = G[LIN({h(r)(x) : r ∈ Q & x ∈ K})] = G[LIN({r + x : r ∈ Q & x ∈ K})] ⊂ G[LIN({r : r ∈ Q})] = G[LIN(Q)].
So, G[LIN(Q)] is dense. It is Gδ by Fact 3.3(d) since LIN(Q) is sigma compact.
We will also need the following fact about perfect sets.
Lemma 3.6. Let G be a proper dense Gδ-subset of R, W a second category Gδ-subset of R, and let M be an Fσ-subset of R such that G[LIN(M )] is a dense Gδ-subset of R. Then there exists a linearly independent set K ∈ Perf(W ) such that G[LIN(M ∪ K)] is dense, LIN(M ) ∩ LIN(K) = { 0 }, and LIN(M ∪ K) \ LIN(M ) ⊂ G. In particular, if M is linearly independent then so is M ∪ K.
Proof. First note that the density of G[LIN(M )] implies LIN(M ) 6 = R. So, LIN(M ) must be of first category. Replacing G with
{q G : q ∈ Q \ { 0 }}, if necessary, we can assume that q G = G for every q ∈ Q \ { 0 }. Notice that then for every q ∈ Q \ { 0 } and linear subspace V of R we also have
q G[V ] = {q x : x−V ⊂ G} = {y : (y/q)−V ⊂ G} = {y : y−q V ⊂ q G} = G[V ].
ON ADDITIVE ALMOST CONTINUOUS FUNCTIONS UNDER CPAgameprism 163
Let J be a non-empty open interval such that W is dense in J and let 〈Gk : k < ω〉 and 〈Wk : k < ω〉 be the decreasing sequences of open subsets of R such that G =
k<ω Gk^ and^ W^ ∩^ J^ =^
k<ω Wk. Choose an increasing sequence 〈Mk : k < ω〉 of compact sets such that LIN(M ) =
k<ω Mk, let^ R be a family of all triples 〈, m, n〉 such that m, n < ω, n > 0, and ∈ Lm+n, where Li’s are as in (3.1), and fix a sequence 〈〈`k, mk, nk〉 ∈ R : k < ω〉 with each triple appearing infinitely many times. We will construct, by induction on k < ω, a sequence 〈Us : s ∈ 2 k^ & k < ω〉 of non-empty open subsets of R such that U∅ = J and for every 0 < k < ω and s ∈ 2 k−^1 the following inductive conditions hold.
(a) cl(Usˆ0) and cl(Usˆ1) are disjoint subsets of Us ∩ Wk. (b) `k(¯a, x 1 ,... , xnk ) ∈ Gk \ Mk for every ¯a ∈ (Mk)mk^ and xj chosen from different Ut with t ∈ 2 k.
To see that such a sequence can be built assume that for some 0 < k < ω the sets {Us : s ∈ 2 k} have been already constructed. Let {ti : i < 2 k} be an enumeration of 2k^ and by induction on i choose
xti ∈ Utik− 1 ∩ (W \ LIN(M ∪ {xtj : j < i}) ∩
y∈LIN{xtj : j ON ADDITIVE ALMOST CONTINUOUS FUNCTIONS UNDER CPAgameprism 165
To make an inductive step assume that for some ξ < ω 1 the required se- quence 〈〈Qζ , R^0 ζ , R^1 ζ 〉 : ζ < ξ〉 is already constructed. So, Rξ is already de- fined and, by the inductive assumption, Rξ is clearly linearly independent. Next notice that
G[LIN(Rξ)] is a dense Gδ.
If ξ = η + 1 then it follows from (v) for η. On the other hand, if ξ is a limit
ordinal then G [LIN(Rξ)] = G
[⋃
η<ξ LIN(Rη+1)
]
η<ξ G^ [LIN(Rη+1)] so it follows from the inductive assumption as well. We define R ξ^0 as a K from Lemma 3.6 applied to W = Bξ and M = Rξ.
This guarantees (iv), Rξ ∩ R^0 ξ = ∅, density of G[LIN(Rξ ∪ R ξ^0 )], and linear
indpendence of Rξ ∪ R^0 ξ.
Next use Lemma 3.5 to prism Pξ and G[LIN(Rξ ∪ R^0 ξ )] to find a subprism Q′^ of Pξ such that
G[LIN(Rξ ∪ R ξ^0 )][LIN(Q′)] = G[LIN(Rξ ∪ R^0 ξ ) + LIN(Q′)] = G[LIN(Rξ ∪ R^0 ξ ∪ Q′)]
is a dense Gδ, where the first equation follows from Fact 3.3(f). Further, apply Lemma 1.5 to M = Rξ ∪ R^0 ξ and prism P = Q′^ to find a subprism Qξ
of Q′^ and a compact R^1 ξ subset of Q′^ \ M such that M ∪ R^1 ξ is a maximal linearly independent subset of M ∪ Qξ. The maximality immediately implies Qξ ⊂ LIN(M ∪ R^1 ξ ) = LIN(Rξ+1) so (iii) holds. We also clearly have (i) and (ii). Condition (v) follows from the density of G[LIN(Rξ ∪ R^0 ξ ∪ Q′)] and the fact that R^1 ξ ⊂ Q′. This finishes the inductive construction. Now, if S is a Player II strategy associated with our construction, then by CPAgameprism, there exists a game 〈〈Pξ , Qξ〉 : ξ < ω 1 〉 played according to S
in which R =
ξ<ω 1 Qξ.^ Let^ 〈〈R
0 ξ , R
1 ξ 〉^ :^ ξ < ω^1 〉^ be a sequence associated with this game. Then H = {Riξ : ξ < ω 1 & i < 2 } is as desired.
Proof of Theorem 3.1. Let X = {C(B) : B ∈ B} where B is as in (3.5). We will play GAMEprism(X ) in which, by Theorem 2.1, Player II has no winning strategy. Notice that since each g ∈ K, where K is defined as in (3.2), contains some function from
X , every function f intersecting each g ∈
X is almost continuous. Let H = {Hξ : ξ < ω 1 } be as in Proposition 3.7. We also fix a sequence P¯ = 〈Pξ : ξ < ω 1 〉 such that each Pξ represents a prism in some C(B) ∈ X. Sequence P¯ represents potential play for Player I in GAMEprism(X ) and we will construct, by induction, a strategy S for Player II which will describe
166 K. CIESIELSKI AND J. PAWLIKOWSKI
a game played according to S in response to P¯. To make S a legitimate strategy its value at stage ξ < ω 1 will depend only on P¯ξ = 〈Pη : η ≤ ξ〉. So, construct a sequence 〈〈H ξ^0 , H ξ^1 , Qξ, Kξ, Rξ, Yξ 〉 : ξ < ω 1 〉 of subsets of R such that for every ξ < ω 1 the following inductive conditions are satisfied, where Bξ ∈ B is such that Pξ ⊂ C(Bξ) and Hξ = {Hηi : η < ξ & i < 2 }. (I) H ξ^0 and H ξ^1 are distinct elements of H \
Hξ. (II) H ξ^0 ∈ [Bξ]c^ and LIN(H ξ^0 ∪
Hξ) \ LIN(
Hξ) ⊂ G.
(III) Hξ ∈ {Hiη : η ≤ ξ & i < 2 }.
We can choose such H ξ^0 and H ξ^1 since H was taken from Proposition 3.7. Also, if Fξ =
η<ξ (Rη^ ∪^ Yη) and^ Uξ^ =^ G[LIN(Fξ)] then (IV) Uξ is a dense Gδ in R, (V) Kξ ∈ Perf(H ξ^0 ), Qξ is a subprism of Pξ, Rξ = {h(x) : h ∈ Qξ, x ∈ Kξ}, and Uξ[LIN(Rξ)] is dense Gδ in R, and (VI) Yξ ∈ Perf(R) is a linearly independent set such that G[LIN(Fξ+1)] is dense, LIN(Fξ ∪ Rξ) ∩ LIN(Yξ) = { 0 }, and LIN(Fξ ∪ Rξ ∪ Yξ) \ LIN(Fξ ∪ Rξ) is a subset of G. Assuming that (IV) holds the possibility of a choice of Qξ, Kξ, and Rξ as in (V) follows directly from Lemma 3.4. Next, since by Fact 3.3(f)
Uξ[LIN(Rξ)] = G[LIN(Fξ)][LIN(Rξ)] = G[LIN(Fξ) + LIN(Rξ)] = G[LIN(Fξ ∪ Rξ )]
we can apply Lemma 3.6 to our G, W = R, and M = Fξ ∪ Rξ to find a linearly independent K ∈ Perf(R) for which G[LIN(Fξ ∪ Rξ ∪ K)] is a dense Gδ subset of R, LIN(Fξ ∪ Rξ ∪ K) \ LIN(Fξ ∪ Rξ) ⊂ G, and LIN(Fξ ∪ Rξ) ∩ LIN(K) = { 0 }. Then put Yξ = K and notice that (VI) is satisfied, since Fξ+1 = Fξ ∪ Rξ ∪ Yξ. To finish the construction it is enough to argue that (IV) is preserved. But if ξ = η + 1 is a successor ordinal then it follows immediately from (VI) for η. But if ξ is a limit ordinal then (IV) follows easily from the density of sets Uη for η < ξ since Uξ = G
[⋃
η<ξ LIN
ζ<η(Rζ^ ∪ {yζ^ })
)]
η<ξ Uη. This finishes the inductive construction of the sequence. We define a strategy S for Player II by S(〈〈Pη, Qη〉 : η < ξ〉, Pξ) = Qξ. By Theorem 2.1 this is not a winning strategy, so there exists a game 〈〈Pξ , Qξ〉 : ξ < ω 1 〉 played according to S in which
X =
ξ<ω 1 Qξ^.^ We will use the sequence 〈〈H^0 ξ , H ξ^1 , Qξ, Kξ, Rξ, yξ 〉 : ξ < ω 1 〉 associated with this game to construct the desired function f. Since, by (I) and (III), {Hξi : ξ < ω 1 & i < 2 } = H, it is enough to define f on each Hξi and extend it to a unique additive function. So, for each ξ < ω
168 K. CIESIELSKI AND J. PAWLIKOWSKI
generalized Erd˝os’ result by proving that, under Martin’s axiom, there exists a Hamel basis H for which E+(H) is simultaneously of measure zero and first category. However, it is unknown whether there is a ZFC example of a Hamel basis H for which E+(H) is of measure zero. In what follows we show that the existence of such a Hamel basis is a consequence of CPAgameprism.
Theorem 4.1. CPAgameprism implies that for every dense Gδ subset G of R with 0 ∈ G there exists an A ⊂ R such that LIN(A) = R and E+(A) ⊂ G.
Using Theorem 4.1 with G of measure zero and the fact that every set A spanning R contains a Hamel basis we obtain immediately the following corollary.
Corollary 4.2. CPAgameprism implies that there exists a Hamel basis H such that E+(H) has measure zero.
Proof of Theorem 4.1. Decreasing G, if necessary, we can assume that qG = G for every non-zero q ∈ Q. Since G has a Polish metric, we can use CPAgameprism for GAMEprism(X) with X = G. Fix a sequence P¯ = 〈Pξ : ξ < ω 1 〉 such that each Pξ represents a prism in X. Sequence P¯ represents a potential play for Player I. We will construct, by induction, a strategy S for Player II which will describe a game played according to S in response to P¯. The value of S at stage ξ < ω 1 will depend only on P¯ξ = 〈Pη : η ≤ ξ〉. For this, we will construct a sequence 〈〈Qξ, Aξ〉 : ξ < ω 1 〉 of pairs of sigma-compact subsets of R such that for every ζ ≤ ξ < ω 1 (I) Qξ is a subprism of Pξ, (II) Aζ ⊂ Aξ and
η≤ξ Qη^ ⊂^ LIN(Aξ), (III) set G[E+(Aξ)] is dense and E+(Aξ) ⊂ G.
Assume that for some ξ < ω 1 the desired sequence 〈〈Qη, Aη〉 : η < ξ〉 is already constructed. Let Bξ =
η<ξ Aη.^ Then^ E
+[Bξ] = ⋃ η<ξ E
+[Aη] is sigma-compact and Gξ = G[E+(Bξ )] =
η<ξ G[E
+(Aη)] is a dense Gδ. Thus, by Lemma 3.5, we can find a subprism Qξ of Pξ such that Gξ[LIN(Qξ)] is a dense Gδ subset of R. Since
Gξ [LIN(Qξ)] = G[E+(Bξ)][LIN(Qξ)] = G[E+(Bξ ) + LIN(Qξ)]
there exists an x ∈ R such that x + E+(Bξ) + LIN(Qξ) ⊂ G. Therefore, we have also qx + E+(Bξ) + LIN(Qξ) ⊂ G for every non-zero q ∈ Q. Let us define Cξ = x + LIN(Qξ) and put Aξ = Bξ ∪ Cξ. This clearly ensures (II). To see E+(Aξ) ⊂ G notice that every element of E+(Aξ ) either belongs to
ON ADDITIVE ALMOST CONTINUOUS FUNCTIONS UNDER CPAgameprism 169
E+(Bξ) ⊂ G or to qx + E+(Bξ) + LIN(Qξ ) ⊂ G for some positive q ∈ Q. The density of G[E+(Aξ)] follows from
G[E+(Aξ )] = G[E+(Bξ) + E+(Cξ)] = G[E+(Bξ)][E+(Cξ)] = Gξ[E+(Cξ )]
= Gξ
[⋃
q∈Q+^ (qx^ + LIN(Qξ^ ))
]
q∈Q+^ Gξ^ [qx^ + LIN(Qξ)]
q∈Q+^ (qx^ +^ Gξ^ [LIN(Qξ)]),
where Q+^ = Q ∩ (0, ∞), since Gξ [LIN(Qξ)] is a dense Gδ. This finishes the inductive construction. Let S be a strategy of Player II given by the above inductive construc- tion. Since S is not winning, there is a game 〈〈Pξ, Qξ〉 : ξ < ω 1 〉 played according to S in which G = X =
ξ<ω 1 Qξ^. Thus, for^ A^ =^
ξ<ω 1 Aξ^ con- dition (III) implies that E+(A) ⊂ G, while by (II) we have R = LIN(G) =
LIN
ξ<ω 1 Qξ
⊂ LIN(A).
References [1] Ciesielski, K., Set Theory for the Working Mathematician, London Math. Soc. Stud. Texts 39 , Cambridge Univ. Press, Cambridge, 1997. [2] Ciesielski, K., Some additive Darboux-like functions, J. Appl. Anal. 4 (1) (1998), 43–
- (Preprint?^ available.^2 ) [3] Ciesielski, K., Jastrz¸ebski, J., Darboux-like functions within the classes of Baire one, Baire two, and additive functions, Topology Appl. 103 (2000), 203–219. (Preprint? available.) [4] Ciesielski, K., Pawlikowski, J., Covering Property Axiom CPAcube and its conse- quences, Fund. Math. 176 (1) (2003), 63–75. (Preprint?^ available.) [5] Ciesielski, K., Pawlikowski, J., Crowded and selective ultrafilters under the Covering Property Axiom, J. Appl. Anal. 9 (1) (2003), 19–55. (Preprint?^ available.) [6] Ciesielski, K., Pawlikowski, J., Uncountable intersections of open sets under CPAprism, Proc. Amer. Math. Soc. 132 (11) (2004), 3379–3385. (Preprint?^ available.) [7] Ciesielski, K., Pawlikowski, J., Small coverings with smooth functions under the Cov- ering Property Axiom, Canad. J. Math. 57 (3) (2005), 471–493. (Preprint?^ available.) [8] Ciesielski, K., Pawlikowski, J., Nice Hamel bases under the Covering Property Axiom, Acta Math. Hungar. 105 (3) (2004), 197–213. (Preprint?^ available.) [9] Ciesielski, K., Pawlikowski, J., Covering Property Axiom CPA. A Combinatorial Core of the Iterated Perfect Set Model, Cambridge Tracts in Math. 164 , Cambridge Univ. Press, Cambridge, 2004. [10] Erd˝os, P., On some properties of Hamel bases, Colloq. Math. 10 (1963), 267–269.
(^2) Preprints marked by? (^) are available in electronic form from Set Theoretic Analysis
Web Page: http://www.math.wvu.edu/homepages/kcies/STA/STA.html
