() @ Applied General Topology c© Universidad Politécnica de Valencia Volume 12, no. 1, 2011 pp. 27-33 Characterizing meager paratopological groups Taras Banakh, Igor Guran and Alex Ravsky Abstract We prove that a Hausdorff paratopological group G is meager if and only if there are a nowhere dense subset A ⊂ G and a countable set C ⊂ G such that CA = G = AC. 2010 MSC: 22A05, 22A30. Keywords: Paratopological group, Baire space, shift-Baire group, shift- meager group. 1. Introduction Trying to find a counterpart of the Lindeföf property in the category of topological groups I.Guran [7] introduced the notion of an ω-bounded group which turned out to be very fruitful in topological algebra, see [9]. We recall that a topological group G is ω-bounded if for each non-empty open subset U ⊂ G there is a countable subset C ⊂ G such that CU = G = U C. A similar approach to the Baire category leads us to the notion of an shift- meager (shift-Baire) group. This is a topological group that can(not) be written as the union of countably many translation copies of some fixed nowhere dense subset. The notion of a shift-meager (shift-Baire) group can be defined in a more general context of semitopological groups, that is, groups G endowed with a shift-invariant topology τ . The latter is equivalent to saying that the group operation · : G × G → G is separately continuous. If this operation is jointly continuous, then (G, τ ) is called a paratopological group, see [1]. A semitopological group G is defined to be • left meager (resp. right meager) if G = CA (resp. G = AC) for some nowhere dense subset A ⊂ X and some countable subset C ⊂ G; • shift-meager if G is both left and right meager; 28 T. Banakh, I. Guran and A. Ravsky • left Baire (resp. right Baire) if for every open dense subset U ⊂ X and every countable subset C ⊂ G the intersection ⋂ x∈C xU (resp. ⋂ x∈C U x) is dense in G; • shift-Baire if G is both left and right Baire. For semitopological groups those notions relate as follows: not shift-Baire HH HH HH HH HY �� �� �� �� �* ? shift-meager @@I ��� �� �� �� �� ��* not left Baire � left meager � ��3 right meager - not right Baire Q QQk HH HH HH HH HHY meager 6 ? not Baire The following theorem implies that for Hausdorff paratopological groups all the eight properties from this diagram are equivalent. Theorem 1.1. A Hausdorff paratopological group G is meager if and only if G is shift-meager. This theorem will be proved in Section 4. The proof is based on Theorem 2.1 giving conditions under which a meager semitopological group is left (right) meager and Theorem 3.2 describing some oscillator properties of 2-saturated Hausdorff paratopological groups. 2. Shift-meager semitopological groups In this section we search for conditions under which a given meager semi- topological group is left (right) meager. Following [3], [4] and [5], [6], we define a subset A ⊂ G of a group G to be • left large (resp. right large) if G = F A (resp. G = AF ) for some finite subset F ⊂ G; • left P-small (resp. right P-small ) if there is an infinite subset B ⊂ G such that the indexed family {bA}b∈B (resp. {Ab}b∈B) is disjoint. Theorem 2.1. A meager semitopological group G is left (right) meager pro- vided one of the following conditions holds: (1) G contains a non-empty open left (right) P-small subset; (2) G contains a sequence (Un)n∈ω of pairwise disjoint open left (right) large subsets; (3) G contains sequences of non-empty open sets (Un)n∈ω and points (gn)n∈ω such that the sets gnUnU −1 n (resp. U −1 n Ungn), n ∈ N, are pairwise dis- joint. Characterizing meager paratopological groups 29 Proof. (1l) Assume that U ⊂ G is a non-empty open left P-small subset. We may assume that U is a neighborhood of the neutral element e of G. It follows that there is a countable subset B = {bn}n∈ω ⊂ G such that bnU ∩ bmU = ∅ for any distinct numbers n 6= m. The countable set B generates a countable subgroup H of G. By an H-cylinder we shall understand an open subset of the form HV g where g ∈ G and V ⊂ U is a neighborhood of e. Let U = {HVαgα : α ∈ A} be a maximal disjoint family of H-cylinders in G (such a family exists by the Zorn Lemma). We claim that ∪U is dense in G. Assuming the converse, we could find a point g ∈ G \ ∪U and a neighborhood V ⊂ U of e such that V g ∩ ∪U. Taking into account that H · (∪U) = ∪U, we conclude that HV g ∩ ∪U = ∅ and hence U ∪{HV g} is a disjoint family of H-cylinders that enlarges the family U, which contradicts the maximality of U. Therefore ∪U is dense in G and hence G \∪U is a closed nowhere dense subset of G. The space G, being meager, can be written as the union G = ⋃ n∈ω Mn of a sequence (Mn)n∈ω of nowhere dense subsets of G. It is easy to see that the set M = (G \ ∪U) ∪ ⋃ α∈A ⋃ n∈ω bn(Mn ∩ Vαgα) is nowhere dense in G and G = HM , witnessing that G is left meager. (2l) Assume that G contains a sequence (Un)n∈ω of pairwise disjoint open left large subsets. For every n ∈ ω find a finite subset Fn ⊂ G with G = Fn ·Un. Write G = ⋃ n∈ω Mn as countable union of nowhere dense subsets and observe that for every n ∈ ω the subset ⋃ x∈Fn x−1(Mn ∩ xUn) of Un is nowhere dense. Since the family {Un}n∈ω is disjoint, the set M = ⋃ n∈ω ⋃ x∈Fn x−1(Mn ∩ xUn) is nowhere dense in G. Since ( ⋃ n∈ω Fn) · M = G, the semitopological group G is left meager. (3l) Assume that (Un)n∈ω is a sequence of non-empty open subsets of G and (gn)n∈ω is a sequence of points of G such that the sets gnUnU −1 n , n ∈ ω, are pairwise disjoint. Using the Zorn Lemma, for every n ∈ ω we can choose a maximal subset Fn ⊂ G such that the indexed family {xUn}x∈Fn is disjoint. If for some n ∈ ω the set Fn is infinite, then the set Un is left P-small and consequently, the group G is left meager by the first item. So, assume that each set Fn, n ∈ ω, is finite. The maximality of Fn implies that for every x ∈ G there is y ∈ Fn such that xUn ∩ yUn 6= ∅. Then x ∈ yUnU −1 n and hence G = FnUnU −1 n , which means that the open set UnU −1 n is left large. Since the family {gnUnU −1 n }n∈ω is disjoint, it is legal to apply the second item to conclude that the group G is left meager. (1r)−(3r). The right versions of the items (1)–(3) can be proved by analogy. � 30 T. Banakh, I. Guran and A. Ravsky 3. Oscillation properties of paratopological groups In this section we establish some oscillation properties of 2-saturated paratopo- logical groups. First, we recall the definition of oscillator topologies on a given paratopological group (G, τ ), see [2] for more details. Given a subset U ⊂ G, by induction define subsets (±U )n and (∓U )n, n ∈ ω, of G letting (±U )0 = (∓U )0 = {e} and (±U )n+1 = U (∓U )n, (∓U )n+1 = U −1(±U )n for n ≥ 0. Thus (±U )n = U U −1U · · · U (−1) n−1 ︸ ︷︷ ︸ n and (∓U )n = U −1U U −1 · · · U (−1) n ︸ ︷︷ ︸ n . Note that ((±U )n)−1 = (±U )n if n is even and ((±U )n)−1 = (∓U )n if n is odd. By an n-oscillator (resp. a mirror n-oscillator ) on a topological group (G, τ ) we understand a set of the form (±U )n (resp. (∓U )n ) for some neighborhood U of the unit of G. Observe that each n-oscillator in a paratopological group (G, τ ) is a mirror n-oscillator in the mirror paratopological group (G, τ −1) and vice versa: each mirror n-oscillator in (G, τ ) is an n-oscillator in (G, τ −1). By the n-oscillator topology on a paratopological group (G, τ ) we understand the topology τn consisting of sets U ⊂ G such that for each x ∈ U there is an n-oscillator (±V )n with x · (±V )n ⊂ U . Let us recall [8], [1, p. 342] that a paratopological group (G, τ ) is saturated if each non-empty open set U ⊂ G has non-empty interior in the mirror topology τ −1 = {U −1 : U ∈ τ}. This notion can be generalized as follows. Define a paratopological group (G, τ ) to be n-saturated if each non-empty open set U ∈ τn has non-empty interior in the topology (τ −1)n. Proposition 3.1. A paratopological group (G, τ ) is 2-saturated if no non-empty open subset U ⊂ G is P-small. Proof. To prove that G is 2-saturated, take any non-empty open set U2 ∈ τ2 and find a point x ∈ U2 and a neighborhood U ∈ τ of e such that xU 2U −2 ⊂ U2. By the Zorn Lemma, there is a maximal subset B ⊂ G such that bU ∩ b′U = ∅ for all distinct points b, b′ ∈ B. By our hypothesis, U is not P-small, which implies that the set B is finite. The maximality of B implies that for each x ∈ G the shift xU meets some shift bU , b ∈ B. Consequently, x ∈ bU U −1 and G = ⋃ b∈B bU U −1. It follows that the closure U U −1 of U U −1 in the topology (τ −1)2 has non-empty interior. We claim that U U −1 ⊂ U 2U −2. Indeed, given any point z ∈ U U −1 , we conclude that the neighborhood U −1zU of z in the topology (τ −1)2 meets U U −1 and hence z ∈ U 2U −2. Now we see that the set U2 ⊃ xU 2U −2 ⊃ xU U −1 has non-empty interior in the topology (τ −1)2, witnessing that the group (G, τ ) is 2-saturated. � Characterizing meager paratopological groups 31 By Proposition 2 of [2], for each saturated paratopological group (G, τ ) the semitopological group (G, τ2) is a topological group. This results generalizes to n-saturated groups. Theorem 3.2. If (G, τ ) is an n-saturated paratopological group for some n ∈ N, then (G, τ2n) is a topological group. Proof. According to Theorem 1 of [2], (G, τ2n) is a topological group if and only if for every neighborhood U ∈ τ of the neutral element e ∈ G there is a neighborhood V ∈ τ of e such that (∓V )2n ⊂ (±U )2n. Since the paratopological group (G, τ ) is n-saturated, the set (±U )n ∈ τ2 contains an interior point x in the mirror topology (τ −1)n. Consequently, there is a neighborhood V ∈ τ of e such that (∓V )nx ⊂ (±U )n. Now we consider separately the cases of odd and even n. 1. If n is odd, then applying the operation of the inversion to (∓V )nx ⊂ (±U )n, we get x−1(±V )n ⊂ (∓U )n and then (∓V )2n = (∓V )n(±V )n = (∓V )nxx−1(±V )n ⊂ (±U )n(∓U )n = (±U )2n. 2. If n is even, then (∓V )nx ⊂ (±U )n implies x−1(∓V )n ⊂ (±U )n and (∓V )2n = (∓V )n(∓V )n = (∓V )nxx−1(∓V )n ⊂ (±U )n(±U )n = (±U )2n. � According to [2], for each 1-saturated Hausdorff paratopological groups (G, τ ) the group (G, τ2) is a Hausdorff topological group. For 2-saturated group we have a bit weaker result. Theorem 3.3. For any non-discrete Hausdorff 2-saturated paratopological group (G, τ ) the maximal antidiscrete subgroup {e} = ⋂ e∈U∈τ (±U )4 of the topological group (G, τ4) is nowhere dense in the topology τ2. Proof. To show that {e} is nowhere dense in the topology τ2, fix any non- empty open set U2 ∈ τ2. Since G is not discrete, so is the topology τ2 ⊂ τ . Consequently, we can find a point x ∈ U2 \ {e}. Since G is a Hausdorff paratopological group, there is a neighborhood U ∈ τ of e such that e /∈ xU U −1 ⊂ U2. The continuity of the group operation yields a neighborhood V ∈ τ of e such that V 2 ⊂ U and V 2x ⊂ xU . Then V 2xV −2 ⊂ xU U −1 6∋ e yields V −1V ∩ V xV −1 = ∅. Using the shift-invariantness of the topology τ , find a neighborhood W ∈ τ of e such that W ⊂ V and xW ⊂ V x. Since the group G is 2-saturated, the open set xW W −1 ∈ τ2 has non-empty interior in the topology (τ −1)2. Consequently, there is a point y ∈ xW W −1 and a neighborhood O ∈ τ of e such that O ⊂ W and O−1yO ⊂ xW W −1. Observe that O−1O ∩ O−1yO ⊂ V −1V ∩ xW W −1 ⊂ V −1V ∩ V xV −1 = ∅ and consequently, U2 ∋ y /∈ OO −1OO−1 ⊃ {e}. � Problem 3.4. Can the topology τ2n be antidiscrete for some Hausdorff n- saturated paratopological group? 32 T. Banakh, I. Guran and A. Ravsky Problem 3.5. Assume that a paratopological group (G, τ ) is 2-saturated. Is its mirror paratopological group (G, τ −1) 2-saturated? 4. Proof of Theorem 1.1 We need to check that each meager Hausdorff paratopological group G is left and right meager. If the paratopological group G contains a non-empty open left P-small sub- set, then G is left meager by Theorem 2.1(1). So assume that no non-empty open subset of G is left P-small. In this case Proposition 3.1 implies that the paratopological group G is 2-saturated while Theorem 3.3 ensures that the topological group (G, τ4) contains a countable disjoint family {Wn}n∈ω of non- empty open sets. By the definition of the 4th oscillator topology τ4 each set Wn contains a subset of the form xnUnU −1UnU −1 n where xn ∈ G and Un is a neighborhood of the neutral element in the paratopological group G. Since the sets xnUnU −1 n ⊂ Wn, n ∈ ω, are pairwise disjoint, we can apply Theorem 2.1(3) to conclude that the paratopological group G is left meager. By analogy we can prove that G is right meager. 5. Discussion and Open Problems The following example shows that without any restrictions, a meager semi- topological group needs not be shift-meager. Example 5.1. Let G be an uncountable group whose cardinality |G| has count- able cofinality. Endow the group G with the shift-invariant topology generated by the base {G \ A : |A| < |G|}. It is easy to see that a subset A ⊂ G is nowhere dense if and only if it is not dense if and only if |A| < |G|. This observation implies that G is meager (because |G| has countable cofinality). On the other hand, the semi-topological group G is not shift-meager because for every nowhere dense subset A ⊂ G and every countable subset C ⊂ G we get |A| < |G| and hence G 6= CA because |CA| ≤ max{ℵ0, |A|} < |G|. Problem 5.2. 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(Received January 2010 – Accepted September 2010) Taras Banakh (T.O.Banakh@gmail.com) Uniwersytet Humanistyczno-Przyrodniczy Jana Kochanowskiego w Kielcach, Poland, and Ivan Franko National University of Lviv, Ukraine Igor Guran (igor guran@yahoo.com) Ivan Franko National University of Lviv, Ukraine Alex Ravsky (oravsky@mail.ru) Institute of Applied Problems of Mechanics and Mathematics of National Academy of Sciences, Lviv, Ukraine Characterizing meager paratopological groups. By T. Banakh, I. Guran and A. Ravsky