Real Analysis Exchange Vol. 27(1), 2001/2002, pp. 707–720

M. Balcerzak^{∗}, Institute of Mathematics, Łódź Technical University, al. Politechniki 11, I-2, 90-924 Łódź, Poland and Faculty of Mathematics, Łódź University, ul. Banacha 22, 90-238 Łódź, Poland. e-mail:

mbalce@krysia.uni.lodz.pl

A. Bartoszewicz, J. Rzepecka and S. Wroński, Institute of Mathematics, Łódź Technical University, al. Politechniki 11, I-2, 90-924 Łódź, Poland.

e-mail: arturbar@ck-sg.p.lodz.pl and wronskis@ck-sg.p.lodz.pl

MARCZEWSKI FIELDS AND IDEALS^{†}

Abstract

For an X ≠ ∅ and a given family F ⊂ P(X) \ {∅}, we consider the Marczewski field S(F) which consists of sets A ⊂ X such that each set U ∈ F contains a set V ∈ F with V ⊂ A or V ∩ A = ∅. We also study the respective ideal S^{0}(F). We show general properties of S(F) and certain representation theorems. For instance we prove that the interval algebra in [0, 1) is a Marczewski field. We are also interested in situations where S(F) = S(τ \ {∅}) for a topology τ on X. We propose a general method which establishes S(F) and S^{0}(F) provided that F is the family of perfect sets with respect to τ, and τ is a certain ideal topology on ℝ connected with measure or category.

1 General properties

708 M. Balcerzak, A. Bartoszewicz, J. Rzepecka and S. Wroński

The notions of (s)-sets and (s^{0})-sets are due to Marczewski [Sz]. They have been investigated by many authors. (See [Mi1], [Mi2], [BrCo] and also [Br2], [Co], [Wa].) The scheme defining (s)-sets and (s^{0})-sets was used for more general settings in several publications (see e.g. [Mo], [Bre], [Pa], [R], [BR]). We observe that this scheme turns out interesting without any essential restrictions on a generating family of sets. Namely, let F be a family of nonempty subsets of a given set X. We put S(F) = {A ⊂ X : (∀ U ∈ F) (∃V ∈ F) (V ⊂ U ∩ A ∨ V ⊂ U \ A)}, S^{0}(F) = {A ⊂ X : (∀ U ∈ F) (∃V ∈ F) V ⊂ U \ A}, H(F) = {A ⊂ X : (∀ B ⊂ A) B ∈ F}. Note that H(F) is the maximal hereditary family contained in F. In the case when F consists of all perfect subsets of a given Polish space, S(F) and S^{0}(F) are exactly the families of classical Marczewski (s)-sets and (s^{0})-sets.

Key Words: Marczewski sets, field of sets, Baire category, Lebesgue measure, density topology.

Mathematical Reviews subject classification: 28A05, 03E75, 06E25, 54A10, 54E52

Received by the editors August 20, 2000

∗The first author was partially supported by NSF Cooperative Research Grant INT-9600548 and its Polish part financed by KBN.

†The main results of the paper were presented during the 13th Spring Miniconference in Real Analysis in Auburn (March 5-6, 1999).

Our notation is standard. By P(X) we denote the power set of X. Throughout the paper X ≠ ∅.

Proposition 1.1. Let F ⊂ P(X) \ {∅}. Then we have

(1) S(F) is a field of sets,

(2) S^{0}(F) ⊂ H(S(F)), and S^{0}(F) is an ideal of sets,

(3) F ∩ S^{0}(F) = ∅,

(4) (∀ U ∈ S(F) \ S^{0}(F))(∃V ∈ F)V ⊂ U,

(5) F ⊂ S(F) ⇔ (∀U, V ∈ F)(∃W ∈ F)(W ⊂ U ∩ V ∨ W ⊂ U \ V),

(6) if {x} ∈ F for all x ∈ X then S(F) = P(X) and S^{0}(F) = {∅}.

Proof. (1) From the definition of S(F) it immediately follows that, if A ∈ S(F) then X \A ∈ S(F). Now, assume that A, B ∈ S(F). Let C ∈ F. If there is a D ∈ F such that either D ⊂ C ∩A or D ⊂ C ∩B then D ⊂ C ∩(A∪B). If such D does not exist, there is a D_{1} ∈ F such that D_{1} ⊂ C \ A and there is a D_{2} ∈ F such that D_{2} ⊂ D_{1} \B. Thus D_{2} ⊂ C \(A∪B). Hence A∪B ∈ S(F).

Similarly, we show that S^{0}(F) is an ideal (condition (2)). The remaining statements, except for (4), can be checked directly without troubles.

(4) Suppose that there is a U_{0} ∈ S(F) \ S^{0}(F) such that V \ U_{0} ≠ ∅ for each V ∈ F. Since U_{0} ∉ S^{0}(F), there is a V_{0} ∈ F such that W ∩ U_{0} ≠ ∅ for each W ∈ F, W ⊂ V_{0}. Since U_{0} ∈ S(F), there is a W_{0} ∈ F such that W_{0} ⊂ V_{0} ∩ U_{0}. But then W_{0} \ U_{0} = ∅, a contradiction.

Corollary 1.1. If F ⊂ P(X) is a field of sets, then

F \ {∅} ⊂ S(F \ {∅}) \ S^{0}(F \ {∅}).

Proof. Use Proposition 1.1 (3) and (5).

Note that the classical (s)-sets and (s^{0})-sets form a σ-field and a σ-ideal, respectively (the proof of σ-additivity for (s^{0})-sets is based on the fusion lemma; see [Sz]). Additionally, in that case H(S(F)) = S^{0}(F) [Sz, 3.1]. On the other hand, there are families F for which H((S(F)) ≠ S^{0}(F) [R, Cor. 1.10]. Observe that there are cases when S(F) forms a σ-field but S^{0}(F) is not a σ-ideal. That happens if X is an infinite set and F stands for the family of all infinite subsets of X; then S(F) = P(X) and S^{0}(F) consists of all finite subsets of X. In [Pa, Lemma 2] it was proved that if F ⊂ S(F) and S^{0}(F) is σ-additive then S(F) is a σ-field.

The operation S can be iterated. For a family F ⊂ P(X) we define S_{0}(F) = F and S_{α}(F) = S(∪_{γ<α}S_{γ}(F)\{∅}) for any ordinal α > 0. Of course we may consider only α ≤ 2^{2^{κ}} where κ is the cardinality of X. The families S_{α}(F), α > 0, are fields and from Corollary 1.1 it follows that S_{γ}(F) ⊂ S_{α}(F) for any ordinals γ, α with 0 < γ < α. In our future studies, we plan to establish the maximal number of different fields that can be obtained in a sequence of type ⟨S_{α}(F): α > 0⟩. In the former version of the paper we claimed incorrectly that this number is 2. The referee has observed that it is at least 3. (See Remark 2.2.)^{1}

We say that two families F_{1}, F_{2} ⊂ P(X) \ {∅} are mutually coinitial if

(∀ U ∈ F_{1}) (∃ V ∈ F_{2}) V ⊂ U
and (∀ U ∈ F_{2}) (∃ V ∈ F_{1}) V ⊂ U.

Proposition 1.2. Let F_{1}, F_{2} ⊂ P(X) \ {∅}.

(i) If F_{1}, F_{2} are mutually coinitial then S(F_{1}) = S(F_{2}) and S^{0}(F_{1}) = S^{0}(F_{2}).

(ii) Assume that F_{1} ⊂ S(F_{1}) and F_{2} ⊂ S(F_{2}). If S(F_{1}) = S(F_{2}) and S^{0}(F_{1}) = S^{0}(F_{2}) then F_{1}, F_{2} are mutually coinitial.

Proof. (i) is evident. (ii) Let U ∈ F_{1}. Then U ∉ S^{0}(F_{1}) by Proposition 1.1(3). Hence U ∈ S(F_{1}) \ S^{0}(F_{1}) = S(F_{2}) \ S^{0}(F_{2}) and by Proposition 1.1(4) there is a V ∈ F_{2} such that V ⊂ U. Analogously, for each U ∈ F_{2} there is a V ∈ F_{1} such that V ⊂ U.

Note that an idea similar to that contained in Proposition 1.2 was used in [Mo, Th. 1, p. 23]. The referee has asked whether the converse of (i) is true. The answer is “no”which follows from Remark 2.1 in the next section.

Now, consider a field Σ (respectively, an ideal I) of subsets of X. We say that:

• Σ (respectively, I) is a topological field (respectively, a topological ideal) if there is a topology τ on X such that Σ consists of all sets with τnowhere dense boundary (respectively, I consists of τ-nowhere dense sets). (Cf. [Ku, §8.V].) We thus write Σ = Σ(τ) and I = NW D(τ).

1S. Wroński has recently proved that this number is exactly 3.

• Σ (respectively, I) is a Marczewski field (respectively, a Marczewski ideal) if there is a family F ⊂ P(X) \ {∅} such that Σ = S(F) (respectively, I = S^{0}(F)).

Note that papers [BET] and [BBC] use different terminology for Marczewski fields: the authors of [BET] say that “F is a basis for a Marczewski-Burstin-like characterization of Σ”, and in [BBC], Σ is called “Marczewski-Burstin representable”.

An easy connection between the above notions is contained in the following

Proposition 1.3. (Cf. [BR].) If τ is a topology on X then S(τ \{∅}) = Σ(τ) and S^{0}(τ \ {∅}) = NW D(τ). Consequently, every topological field (ideal) is a Marczewski field (ideal).

From Proposition 1.2(i) we derive

Proposition 1.4. If a family F ⊂ P(X) \ {∅} is mutually coinitial with a topological base on X then the field S(F) is topological.

In Section 2 we shall show that the interval algebra is a Marczewski field. This will imply that there are Marczewski fields that are not topological fields. A Boolean-theoretical characterization of topological fields was given in [Wr1]. Article [CJ] was devoted to extensive studies of topological ideals; the authors considered also an additional requirement stating that an ideal consists of meager sets in some topology. In Section 2 we discuss some connections between Marczewski fields, topological fields and category bases (introduced by John Morgan II, see [Mo]).

The class of Marczewski fields seems to be rich. From [Bu] it follows that the Lebesgue measurable sets in ℝ form a Marczewski field. (Note that paper [Bu] is much earlier than [Sz].) Also the sets with the Baire property in ℝ constitute a Marczewski field [Br1], [BET]. When we started to prepare our paper, it was not even known whether there exists a non-Marczewski field of subsets of ℝ. Now, our knowledge is wider. Namely, the forthcoming paper [BBC] contains a construction of a non-Marczewski field on ℝ provided 2^{ω} = ω_{1} and 2^{ω_{1}} = ω_{2}. Another result of [BBC] states that 2^{ω} = ω_{1} and 2^{ω_{1}} = ω_{2} imply that the Borel subsets of ℝ form a Marczewski field.

For any filter F of the algebra A = P(X) we denote −F = {X\E : E ∈ F} and A_{F} = F ∪ −F.

Proposition 1.5. For any filter F of A we have S(F) = A_{F} and S^{0}(F) = −F.

Proof. We easily check that F ⊂ S(F) and −F ⊂ S^{0}(F). Since A_{F} is the smallest field containing F, we have A_{F} ⊂ S(F). To show the reverse inclusion consider a U ∈ S(F). Since X ∈ F, we can find a V ∈ F such that either V ⊂ U or V ⊂ X \ U. Hence either U ∈ F or X \ U ∈ F which means that U ∈ A_{F}. Thus S(F) ⊂ A_{F}. It can be similarly shown that S^{0}(F) ⊂ −F.

Proposition 1.6. For a set X of cardinality |X| = κ there are 2^{2^{κ}} nonisomorphic Marczewski fields on X containing all singletons.

Proof. We follow the argument given in [F]. Let Φ consist of all filters in A = P(X) which are intersections of two free ultrafilters. Then |Φ| = 2^{2^{κ}} and A_{F_{1}} ≠ A_{F_{2}} for any distinct F_{1}, F_{2} ∈ Φ. Additionally, {x} ∈ A_{F} for any x ∈ X and F ∈ Φ. Thus, by Proposition 1.5, there are 2^{2^{κ}} Marczewski fields on X containing all singletons. Any isomorphism between subalgebras of P(X) containing all singletons is induced by a bijection from X to X. Hence each isomorphism class of such subalgebras has at most 2^{κ} elements. Finally, observe that if h is a bijection of X onto X and F ∈ Φ then

{h[U] : U ∈ S(F)} = S({h[V] : V ∈ F}) and {h[V] : V ∈ F} ∈ Φ.

Thus there are 2^{2^{κ}} different classes of isomorphic Marczewski fields on X containing all singletons.

2 Marczewski fields, topological fields and category bases

Let X = [0, 1). The family of all finite unions of half-open intervals [a, b) (where 0 ≤ a < b ≤ 1) form a field of subsets of X. It is called the interval algebra of X [K, 1.11].

Theorem 2.1. The interval algebra A of X = [0, 1) is a Marczewski field.

Proof. Let ℚ stand for the set of all rationals and let 𝔠 denote the cardinality of ℝ. Consider the equivalence relation x ∼ y ⟺ x−y ∈ ℚ. Let F: [0, 1] → ℝ/ ∼ be a one-to-one function such that x ∉ F (x) for x ∈ [0, 1]. (Note that F can be easily constructed by transfinite induction. Indeed, arrange all points of [0, 1] into a one-to-one sequence x_{γ}, γ < 𝔠, and consider an α < 𝔠. If the values F (x_{γ}) for γ < α have been defined, we pick x ∈ [0, 1] \ ⋃_{γ<α}[F (x_{γ})] \ [x_{α}] and put F (x_{α}) = [x] where [x] denotes the respective equivalence class.)

For x ∈ [0, 1] let F_{r}(x) = {([x, x + ε) \ F (x)) ∩ X : ε > 0}, F_{l}(x) = {((x − ε, x) \ F (x)) ∩ X : ε > 0}, F(x) = F_{l}(x) ∪ F_{r}(x). Note that F_{r}(1) = F_{l}(0) = ∅, otherwise F_{r}(x) and F_{l}(x) are nonempty. Finally, let F = ⋃_{x∈[0,1]} F(x).

From the definitions of F and S(F) it easily follows that [a, b) ∈ S(F) for any a, b with 0 ≤ a < b ≤ 1. Since S(F) is a field of sets, we have A ⊂ S(F).

Claim 1. Let x ∈ [0, 1] and k ∈ {r, l}. If U ∈ F_{k}(x) then for each y ∈ [0, 1] and for each V ∈ F(y) such that V ⊂ U, we have y = x, and moreover V ∈ F_{k}(x).

Indeed, suppose that y ≠ x. Let U = I \ F (x) and V = J \ F (y) where I and J are the respective intervals. From V ⊂ U and the density of F (x) it follows that ∅ ≠ J ∩ F (x) ⊂ J ∩ F (y) which contradicts the disjointness of F (x) and F (y). Thus y = x and so, V ∈ F_{m}(x) for some m ∈ {r, l}. However, m = k since otherwise U ∩ V = ∅.

We have already observed that A ⊂ S(F). To prove that S(F) ⊂ A fix an A ∈ S(F) \ {∅}.

Claim 2. For each x ∈ A there exists an ε > 0 such that [x, x + ε) ⊂ A. For each x ∈ X \ A there exists an ε > 0 such that [x, x + ε) ∩ A = ∅.

The latter assertion follows from the former applied to X \ A. To show the former assertion, suppose x ∈ A and [x, x + ε) \ A ≠ ∅ for each ε > 0. Consider a U ∈ F_{r}(x). Since A ∈ S(F), there is a V ∈ F such that either V ⊂ U ∩ A or V ⊂ U \ A. By Claim 1 we have V ∈ F_{r}(x). Since x ∈ A ∩ V, we infer that V ⊂ U ∩ A. Let V = ([x, x + ε) \ F (x)) ∩ X where ε > 0. We may assume that x + ε ≤ 1. By our supposition, pick a y ∈ (x, x + ε) \ A. Let Ṽ = [y, x + ε) \ F (y). Then Ṽ ∈ F_{r}(y) and since A ∈ S(F), there is a W ∈ F such that either W ⊂ Ṽ ∩A or W ⊂ Ṽ \A. Again, by Claim 1, we have W ∈ F_{r}(y), so we may assume that W = [y, y+ε_{1})\F (y) where y+ε_{1} ≤ x+ε. Since y ∉ A, we have W ∩A = ∅. The set [y, y+ε_{1})\(F (x)∪F (y)) is nonempty (uncountable) contained in [x, x + ε) \ F (x) = V ⊂ A and simultaneously in [y, y + ε_{1}) \ F (y) = W ⊂ X \ A. Contradiction.

Claim 3. For each x ∈ (0, 1] there exists an ε > 0 such that either (x−ε, x) ⊂ A or (x − ε, x) ∩ A = ∅.

To show the claim, suppose that there exists an x ∈ (0, 1] such that (x − ε, x) \ A ≠ ∅ and (x − ε, x) ∩ A ≠ ∅ for each ε > 0. Let U ∈ F_{l}(x). Since A ∈ S(F), there is a V ∈ F such that either V ⊂ U ∩ A or V ⊂ U \ A. By Claim 1 we have V ∈ F_{l}(x) and we may assume that V = (x − ε, x) \ F (x) where x−ε ≥ 0. If V ⊂ A, by our supposition we can pick y ∈ (x−ε, x)\A. By Claim 2 there is an ε_{1} > 0 such that [y, y+ε_{1})∩A = ∅ and we may assume that y+ε_{1} ≤ x. On the other hand, ∅ ≠ [y, y+ε_{1})\F (x) ⊂ (x−ε, x)\F (x) ⊂ V ⊂ A, a contradiction. If V ∩A = ∅, by our supposition we can pick y ∈ (x−ε, x)∩A. By Claim 2 there is an ε_{1} > 0 such that [y, y + ε_{1}) ⊂ A and we may assume that y +ε_{1} ≤ x. On the other hand, ∅ ≠ [y, y +ε_{1})\F (x) ⊂ (x−ε, x)\F (x) = V ⊂ X \ A, a contradiction.

From Claim 2 it follows that each connected component I of A is a nondegenerate interval with b = sup I ∉ I. Denote a = inf I and observe that a ∈ I. Indeed, suppose that a ∉ I. We know that [a, b) ∈ S(F). Thus A ∩ [a, b) = (a, b) ∈ S(F) and consequently [a, b) \ (a, b) = {a} ∈ S(F) which contradicts Claim 2.

From the above we infer that A is a union of at most countable family of pairwise disjoint intervals of type [a, b). This family however cannot be infinite. Indeed, suppose that A = ⋃_{n=1}^{∞}[a_{n}, b_{n}) with [a_{n}, b_{n}) ⊂ X, n ≥ 1, pairwise disjoint. Pick a strictly monotonic subsequence (a_{k_{n}}) of (a_{n}). If a_{k_{n}} ↘ x, we apply Claim 2 to x and we obtain a contradiction. If a_{k_{n}} ↗ x, we apply Claim 3 to x and we obtain a contradiction.

Thus we have proved that A ∈ A. Consequently, S(F) ⊂ A.

Remark 2.1. Observe that in the above construction, we can choose, for i = 1, 2, one-to-one functions F^{(i)} : [0, 1] → ℝ/ ∼ with disjoint ranges, and such that x ∉ F^{(i)}(x) for each x ∈ [0, 1]. Then A = S(F^{(1)}) = S(F^{(2)}) where F^{(i)} (i = 1, 2) is associated with F^{(i)} as in the proof of Theorem 2.1. Since H(A) = {∅}, we have S^{0}(F^{(1)}) = S^{0}(F^{(2)}) = {∅} by Proposition 1.1(2). However, the argument for Claim 1 shows that F^{(1)} and F^{(2)} are not mutually coinitial. Thus the converse of (i) in Proposition 1.2 is false.

Remark 2.2. Since A\{∅} and the family of nonempty open sets in [0, 1) are mutually coinitial, the field Σ = S(A \ {∅}) consists of all sets in [0, 1) with nowhere dense boundary, and S(Σ\{∅}) = P([0, 1)) by Proposition 1.1(6). So we have 3 different fields obtained by the iteration of S(·).

Corollary 2.1. There exists an Marczewski field which is not a topological field.

Proof. This follows from Theorem 2.1 since every topological field has an atom [Wr1] and the algebra A has no atoms.

Although the class of topological subfields of P(X) is smaller than the class of Marczewski fields, the former can be used to get the following representation result:

Theorem 2.2. Every field Σ of subsets of X is equal to the intersection of all topological fields containing Σ.

Proof. If Σ = P(X), the assertion is obvious. Assume that Σ ≠ P(X). It suffices to show that for each A ∉ Σ there is a topological field Σ_{A} ⊃ Σ with A ∉ Σ_{A}. So, let A ∉ Σ. By [Wr2, Lemma 2] we find an ultrafilter F_{A} of the field Σ such that no subset of A is in F_{A} and no subset of X \A is in F_{A}. Thus A ∉ S(F_{A}). Put Σ_{A} = S(F_{A}). Observe that Σ_{A} is a topological field since F_{A} forms a topological base, and thus Proposition 1.4 can be used. Because F_{A} is an ultrafilter of Σ, we have Σ ⊂ S(F_{A}).

Recall [Mo] that a pair (X, C) is said to be a category base if C stands for a family of subsets of a nonempty set X, and nonempty sets in C, called regions, satisfy the following axioms:

1^{0} ⋃ C = X,

2^{0} Let A be a region and D – a nonempty family of disjoint regions with |D| < |C|. Then

• if A ∩ (⋃ D) contains a region then there is a region D ∈ D such that A ∩ D contains a region;

• if A ∩ (⋃ D) contains no region then there is a region B ⊂ A \ ⋃ D.

A set E ⊂ X is called singular if E ∈ S^{0}(C \ {∅}).

Theorem 2.3. Assume that for a family F ⊂ P(X) \ {∅} we have X ∈ F ⊂ S(F) and ⋃(S^{0}(F) ∩ {U ∩ V : V ∈ F}) ∈ S^{0}(F) for each U ∈ F. Then (X, F) is a category base whose ideal of singular sets equals S^{0}(F).

Proof. It is enough to check condition 2^{0} defining a category base. Let F, F_{t} ∈ F for t ∈ T (where T is an arbitrary set of indices). Assume that F ∩ ⋃_{t∈T} F_{t} contains a set from F. Then there exists a t_{0} ∈ T such that F ∩ F_{t_{0}}

contains a set from F. Indeed, suppose that it is not the case. Since F ⊂ S(F), we have F ∩ F_{t} ∈ S(F) for each t ∈ T. Hence by Proposition 1.1(4) we get F ∩ F_{t} ∈ S^{0}(F). Thus, by assumption, we have ⋃_{t∈T}(F ∩ F_{t}) ∈ S^{0}(F) which

yields a contradiction (cf. Proposition 1.1(3)). Assume now that F ∩ ⋃_{t∈T} F_{t}

contains no set from F. Then every set F ∩ F_{t}, t ∈ T, contains no set from F. As before we infer that F ∩F_{t} ∈ S(F) for t ∈ T and moreover F ∩F_{t} ∈ S^{0}(F). By assumption we have ⋃_{t∈T}(F ∩ F_{t}) ∈ S^{0}(F) and consequently,

F \ ⋃_{t∈T}(F ∩ F_{t}) ∈ S(F) \ S^{0}(F),

since F ∈ S(F) \ S^{0}(F). Hence, by Proposition 1.1(4), there is a set from F contained in F \ ⋃_{t∈T} F_{t}.

3 Marczewski fields and perfect sets

Let (X, τ) be a topological space. By a τ-perfect set we mean a nonempty τ-closed set without isolated points. Let Perf(τ) stand for the family of all τ-perfect subsets of X. As it was mentioned in Section 1, if X is a Polish space with topology τ then S(Perf(τ)) and S^{0}(Perf(τ)) are exactly the classical families of Marczewski (s)-sets and (s^{0})-sets. In [R], studies of S(Perf(τ)) and S^{0}(Perf(τ)) for other topological spaces were initiated. (See also [BR].) In this section we propose a general method which enables us to reprove some results of [R] and [BR], and to show new applications.

If τ is a given topology on X and we want to characterize S(Perf(τ)), we shall use the following scheme:

1^{0} we conjecture that S(Perf(τ)) = Σ where Σ is a known field of sets,

2^{0} we know that Σ = S(F) for some family F ⊂ P(X) \ {∅};

3^{0} in aim to confirm our conjecture 1^{0}, it is enough (by Proposition 1.2(i)) to check that Perf(τ) and F are mutually coinitial.

A similar method works for S^{0}(Perf(τ)). The above scheme will be illustrated by examples dealing with some “ideal topologies” on ℝ.

Let τ be a topology on X, and let I ⊂ P(X) be a σ-ideal containing all singletons. The family

B_{I}^{⋆} = {U \ A : U ∈ τ & A ∈ I}

forms a base for a topology τ_{I}^{⋆}, on X, stronger than τ, which will be called the Hashimoto topology associated with τ and I. If τ is second countable (or even hereditary Lindelöf) then τ_{I}^{⋆} = B_{I}^{⋆}. (See [H], [JH], [LMZ].) The following property is well known.

Lemma 3.1. (Cf. [BR]). Let I be a σ-ideal of subsets of a separable metric space X and let I contain all singletons. A set F ⊂ X is τ_{I}^{⋆}-perfect if and only if F is τ-perfect and U ∩ F ∉ I for each U ∈ τ with U ∩ F ≠ ∅.

By M and N we denote, respectively, the σ-ideals all meager (i.e. of the first category) sets and of all Lebesgue null sets in ℝ. We shall consider the Hashimoto topologies T_{M}^{⋆} and T_{N}^{⋆} where T stands for the natural topology on ℝ. Let D_{N} denote the density topology on ℝ. (See e.g. [O] or [CLO].) Wilczyński in [W1] introduced the category analogue of the density topology which will be denoted by the D_{M}. Since topology D_{M} is less known, let us give necessary definitions. A number x ∈ ℝ is called a category density point of a set A ⊂ ℝ with the Baire property if each increasing sequence {n_{k}} of positive integers has a subsequence {n_{m_{k}}} such that the sequence of characteristic functions

χ_{[−1,1]∩n_{m_{k}} (A−x)}(t)

(where n_{m_{k}}(A − x) = {n_{m_{k}}(a − x) : a ∈ A}) tends to χ_{[−1,1]}(t) for all points t ∈ ℝ except for those belonging to a meager set. If [−1, 1] is replaced by [−1, 0] or [0, 1], we get the respective notions of one-sided category density points. Topology D_{M} consists of all sets A ⊂ ℝ with the Baire property, such that each point of A is a category density point of A. There are many analogies between D_{M} and D_{N}; for details, see [PWW], [W2] and [CLO].

It is known that the only possible inclusions between the above-mentioned topologies are the following T ⊊ T_{M}^{⋆} ⊊ D_{M} and T ⊊ T_{N}^{⋆} ⊊ D_{N}. (See [LJW].)

The following proposition is due to W. Wilczyński (oral communication).

Proposition 3.1. Every D_{M}-perfect set has nonempty T -interior.

Proof. Let F be a D_{M}-perfect set. Then F is nonmeager. Indeed, if F ∈ M and x ∈ F then U = (ℝ \ F) ∪ {x} ∈ D_{M} and F ∩ U = {x} which contradicts the fact that F is a D_{M}-perfect set. Since F is a nonmeager set with the Baire property, there is an open interval V such that F ∩ V is comeager in V. We claim that V ⊂ F. Let x ∈ V and let W be a D_{M}-neighborhood of x. Thus x is a category density point of the both sets V and W. Consequently, V ∩ W ∉ M. Since F ∩ V is comeager in V, we thus have F ∩ W ≠ ∅. Hence x belongs to the the D_{M}-closure of F (equal to F).

Let Σ_{M} and Σ_{N} denote, respectively, the σ-fields of all sets with the Baire property and of all Lebesgue measurable sets in ℝ. As in Section 1, Σ(T) stands for the field of all subsets of ℝ with nowhere dense boundary, and NW D(T) - the ideal of nowhere dense subsets of ℝ.

Theorem 3.1. (a) S(P erf(D_{N})) = Σ_{N}, S^{0}(P erf(D_{N})) = N. (See [R].)

(b) S(P erf(T_{N}^{⋆})) = Σ_{N}, S^{0}(P erf(T_{N}^{⋆})) = N. (See [BR].)

(c) S(P erf(T_{M}^{⋆})) = Σ(T), S^{0}(P erf(T_{M}^{⋆})) = NW D(T). (See [BR].)

(d) S(P erf(D_{M})) = Σ(T), S^{0}(P erf(D_{M})) = NW D(T).

Proof. (a) Let F denote the family of all T -perfect sets of positive measure. Burstin [Bu] proved that Σ_{N} = S(F). It is not hard to prove that F and Perf(D_{N}) are mutually coinitial (cf. [R, Lemma 3.2]). Thus by our scheme we have S(Perf(D_{N})) = Σ_{N}. Moreover S^{0}(F) ⊂ H(S(F)) = H(Σ_{N}) = N and N ⊂ Σ_{N} = S(F). But N ⊂ S(F) easily implies that N ⊂ S^{0}(F). Hence N = S^{0}(F) = S^{0}(Perf(D_{N})).

(b) If F is as in (a), Lemma 3.1 easily implies that F and Perf(T_{N}^{⋆}) are mutually coinitial (cf. [BR]). The rest is the same as in (a).

(c) From Lemma 3.1 it follows that T_{M}^{⋆}-perfect sets have nonempty T - interior. Also, a nonempty T -open set contains a T_{M}^{⋆}-perfect set (a closed nondegenerate interval). Hence Perf(T_{M}^{⋆}) and T \ {∅} are mutually coinitial. Thus the assertion follows from Proposition 1.3.

(d) As in (c) it suffices to prove that Perf(D_{M}) and T \ {∅} are mutually coinitial. Firstly, by Proposition 3.1, every D_{M}-perfect set has nonempty T -interior. Secondly, let us show that a nondegenerate interval [a, b] is a D_{M}-perfect set. Indeed, [a, b] is obviously D_{M}-closed. Let x ∈ [a, b] and x ∈ U ∈ D_{M}. Then x is a category density point of U and x is at least a one-sided category density point of [a, b]. Hence U ∩ [a, b] ∉ M and so, x is a D_{M}-accumulation point of [a, b].

Note that statements (a) and (b),(c) of Theorem 3.1 can be extended to cases dealing with spaces more general than ℝ, as it was mentioned in [R] and [BR]. Topology D_{M} can be considered in certain linear topological spaces and statement (d) of Theorem 3.1 then holds.

Assertions (a),(d) and (b),(c) of Theorem 3.1 show a kind of asymmetry between measure and category. Knowing (a),(b) we rather expected to obtain Σ_{M} and M as the respective Marczewski families in (c),(d). So the following problem appears:

Problem 3.1. Find a topology τ on ℝ such that Σ_{M} (the σ-field of all subsets of ℝ with the Baire property) is of the form S(Perf(τ)).

Note that Brown [Br1] (see also [BET]) showed the equalities Σ_{M} = S(G) and M = S^{0}(G) provided that G consists of sets of the form U \ F where U is open and F is an F_{σ} meager set. This easily implies that Σ_{M} = S(T_{M}^{⋆}) and M = S^{0}(T_{M}^{⋆}) since G and T_{M}^{⋆} are mutually coinitial. Thus Σ_{M} is a topological field.

Acknowledgements. We would like to thank the referee for several valuable remarks.

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