On subgroups of saturated or totally bounded paratopological groups
aa r X i v : . [ m a t h . G R ] M a r ON SUBGROUPS OF SATURATED OR TOTALLY BOUNDEDPARATOPOLOGICAL GROUPS
TARAS BANAKH AND SASHA RAVSKY
Abstract.
A paratopological group G is saturated if the inverse U − of each non-emptyset U ⊂ G has non-empty interior. It is shown that a [first-countable] paratopologicalgroup H is a closed subgroup of a saturated (totally bounded) [abelian] paratopologicalgroup if and only if H admits a continuous bijective homomorphism onto a (totallybounded) [abelian] topological group G [such that for each neighborhood U ⊂ H of theunit e there is a closed subset F ⊂ G with e ∈ h − ( F ) ⊂ U ]. As an application weconstruct a paratopological group whose character exceeds its π -weight as well as thecharacter of its group reflexion. Also we present several examples of (para)topologicalgroups which are subgroups of totally bounded paratopological groups but fail to besubgroups of regular totally bounded paratopological groups. In this paper we continue investigations of paratopological groups, started by the au-thors in [Ra ], [Ra ], [Ra ], [BR ]–[BR ]. By a paratopological group we understand apair ( G, τ ) consisting of a group G and a topology τ on G making the group operation · : G × G → G of G continuous (such a topology τ will be called a semigroup topology on G ). If, in addition, the operation ( · ) − : G → G of taking the inverse is continuouswith respect to the topology τ , then ( G, τ ) is a topological group . All topological spacesconsidered in this paper are supposed to be Hausdorff if the opposite is not stated.The absence of the continuity of the inverse in paratopological groups results in appear-ing various pathologies impossible in the category of topological groups, which makes thetheory of paratopological groups quite interesting and unpredictable. In [Gu] I. Guran hasintroduced a relatively narrow class of so-called saturated paratopological groups whichbehave much like usual topological groups. Following I.Guran we define a paratopolog-ical group G to be saturated if the inverse U − of any nonempty open subset U of G has non-empty interior in G . A standard example of a saturated paratopological groupwith discontinuous inverse is the Sorgenfrey line , that is the real line endowed with theSorgenfrey topology generated by the base consisting of half-intervals [ a, b ), a < b . Im-portant examples of saturated paratopological groups are totally bounded groups, that isparatopological groups G such that for any neighborhood U ⊂ G of the origin in G thereis a finite subset F ⊂ G with G = F U = U F , see [Ra , Proposition 3.1].Observing that each subgroup of the Sorgenfrey line is saturated, I.Guran asked if thesame is true for any saturated paratopological group. This question can be also posed inanother way: which paratopological groups embed into saturated ones? A similar questionconcerning embedding into totally bounded semi- or paratopological groups appeared in[AH].In this paper we shall show that the necessary and sufficient condition for a paratopo-logical group G to embed into a saturated (totally bounded) paratopological group is Mathematics Subject Classification.
Key words and phrases. saturated paratopological group, group reflexion.The first author was supported in part by the Slovenian-Ukrainian research grant SLO-UKR 02-03/04. the existence of a bijective continuous group homomorphism h : G → H onto a topo-logical (totally bounded) group H . The latter property of G will be referred to as the ♭ -separateness (resp. Bohr separateness). ♭ -Separated paratopological groups can be equivalently defined with help of the groupreflexion of a paratopological group. Given a paratopological group G let τ ♭ be thestrongest group topology on G , weaker than the topology of G . The topological group G ♭ = ( G, τ ♭ ), called the group reflexion of G , has the following characteristic property:the identity map i : G → G ♭ is continuous and for every continuous group homomorphism h : G → H from G into a topological group H the homomorphism h ◦ i − : G ♭ → H iscontinuous. According to [BR ], a neighborhood base of the unit of the group reflexion G ♭ of a saturated (more generally, 2-oscillating) paratopological group G consists of thesets U U − where U runs over neighborhoods of the unit in G . For instance, the groupreflexion of the Sorgenfrey line is the usual real line.There is also a dual notion of a group co-reflexion. Given a paratopological group G let τ ♯ be the weakest group topology on G , stronger than the topology of G . Thetopological group G ♯ = ( G, τ ♯ ) is called the group co-reflexion of G . According to [Ra ],a neighborhood base of the unit of the group co-reflexion G ♯ of a paratopological group G consists of the sets U ∩ U − where U runs over neighborhoods of the unit in G . Aparatopological group is called ♯ -discrete provided its group co-reflection is discrete. Forinstance, the Sorgenfrey line is ♯ -discrete.A subset A of a paratopological group G will be called ♭ -closed if A is closed in thetopology τ ♭ . A paratopological group G is called ♭ -separated provided its group reflexion G ♭ is Hausdorff; G is called ♭ -regular if each neighborhood U of the unit e of G containsa ♭ -closed neighborhood of e . It is easy to see that each ♭ -regular paratopological groupis regular and ♭ -separated, see [BR ]. For saturated groups the converse is also true: eachregular saturated paratopological group is ♭ -regular, see [BR , Theorem 3].The notions of a ♭ -separated (resp. ♭ -regular) paratopological group is a partial case ofa more general notion of a G -separated (resp. G -regular) paratopological group, where G is a class of topological groups. Namely, we call a paratopological group G to be • G -separated if G admits a continuous bijective homomorphism h : G → H onto atopological group H ∈ G ; • G -regular if G admits a regular continuous homomorphism h : G → H onto atopological group H ∈ G .A continuous map h : X → Y between topological spaces is defined to be regular if foreach point x ∈ X and a neighborhood U of x in X there is a closed subset F ⊂ Y suchthat h − ( F ) is a closed neighborhood of x with h − ( F ) ⊂ U .As we shall see later, any injective continuous map from a k ω -space is regular. Weremind that a topological space X is defined to be a k ω -space if there is a countable cover K of X by compact subsets of X , determining the topology of X in the sense that a subset U of X is open in X if and only if the intersection U ∩ K is open in K for any compact set K ∈ K . According to [FT], each Hausdorff k ω -space is normal. Under a (para)topological k ω -group we shall understand a (para)topological group whose underlying topologicalspace is k ω . Many examples of k ω -spaces appear in topological algebra as free objects invarious categories, see [Cho]. In particular, the free (abelian) topological group over acompact Hausdorff space is a topological k ω -group, see [Gra]. N SUBGROUPS OF SATURATED PARATOPOLOGICAL GROUPS 3
Proposition 1.
Any injective continuous map f : X → Y from a Hausdorff k ω -space X into a Hausdorff space Y is regular.Proof. Fix any point x ∈ X and an open neighborhood U ⊂ X of x . Let { K n } be anincreasing sequence of compacta determining the topology of the space X . Without lossof generality, we may assume that K = { x } . Let V = K and W = Y .By induction, for every n ≥ V n of x in K n andan open neighborhood W n of f ( V n ) in Y such that1) f − ( W n ) ∩ K n ⊂ U ;2) V n ⊂ U ∩ T i ≤ n f − ( W i );3) V n ⊂ V n +1 .Assume that for some n the sets V i , W i , i < n , have been constructed. It followsfrom (2) that f ( V n − ) and f ( K n \ U ) are disjoint compact sets in the Hausdorff space Y . Consequently, the compact set f ( V n − ) has an open neighborhood W n ⊂ Y whoseclosure W n misses the compact set f ( K n \ U ). Such a choice of W n and the condition(2) imply that V n − ⊂ U ∩ T i ≤ n f − ( W i ). Using the normality of the compact space K n find an open neighborhood V n of V n − in K n such that V n ⊂ U ∩ T i ≤ n f − ( W i ), whichfinishes the inductive step.It is easy to see that V = S n ∈ ω V n is an open neighborhood of x such that f − ( f ( V )) ⊂ f − ( T i ∈ ω W i ) ⊂ U , which just yields the regularity of the map f . (cid:3) For paratopological k ω -groups this proposition yields the equivalence between the G -regularity and G -separateness. Corollary 1.
Let G be a class of topological groups. A paratopological k ω -group is G -separated if and only if it is G -regular. (cid:3) A class G of topological groups will be called • closed-hereditary if each closed subgroup of a group G ∈ G belongs to the class G ; • H -stable , where H is a topological group, if G × H ∈ G for any topological group G ∈ G .For a topological space X by χ ( X ) we denote its character , equal to the smallestcardinal κ such that each point x ∈ X has a neighborhood base of size ≤ κ .Now we are able to give a characterization of subgroups of saturated paratopologicalgroups (possessing certain additional properties). Theorem 1.
Suppose T is a saturated ♯ -discrete nondiscrete paratopological group and G is a closed-hereditary T ♭ -stable class of topological groups. A paratopological group H is G -separated if and only if H is a ♭ -closed subgroup of a saturated paratopological group G with G ♭ ∈ G , χ ( G ) = max { χ ( H ) , χ ( T ) } , and | G/H | = | T | . A similar characterization holds for G -regular paratopological groups. We remind thata paratopological group G is called a paratopological SIN-group if any neighborhood U ofthe unit e in G contains a neighborhood W ⊂ G of e such that gW g − ⊂ U for all g ∈ G .It is easy to check that every paratopological SIN-group has a base at the unit consistingof invariant open neighborhoods, see [Ra , Ch.4] (as expected, a subset A of a group G iscalled invariant if xAx − = A for all x ∈ G ).Finally we define a notion of a Sorgenfrey paratopological group which crystallizes someimportant properties of the Sorgenfrey topology on the real line. A paratopological group G is defined to be Sorgenfrey if G is non-discrete, saturated and contains a neighborhood TARAS BANAKH AND SASHA RAVSKY U of the unit e such that for any neighborhood V ⊂ G of e there is a neighborhood W ⊂ G of e such that x, y ∈ V for any elements x, y ∈ U with xy ∈ W . Observe thateach Sorgenfrey paratopological group is ♯ -discrete. Theorem 2.
Suppose T is a first countable saturated regular Sorgenfrey paratopologicalSIN-group and G is a closed-hereditary T ♭ -stable class of first countable topological SIN-groups. A paratopological group H is G -regular if and only if H is a ♭ -closed subgroup of asaturated ♭ -regular paratopological group G with G ♭ ∈ G , χ ( G ) = χ ( H ) , and | G/H | = | T | . Theorem 2 in combination with Corollary 1 yield
Corollary 2.
Suppose T is a first countable saturated regular Sorgenfrey paratopologicalSIN-group and G is a closed-hereditary T ♭ -stable class of first countable topological SIN-groups. A paratopological k ω -group H is G -separated if and only if H is a ♭ -closed subgroupof a saturated ♭ -regular paratopological group G with G ♭ ∈ G , χ ( G ) = χ ( H ) , and | G/H | = | T | . (cid:3) As we said, any (regular) saturated paratopological group is ♭ -separated (and ♭ -regular),see [BR ]. Observe that a paratopological group G is ♭ -separated if and only if G is TopGr-separated where TopGr stands for the class of all Hausdorff topological groups. Theseobservations and Theorem 1 imply Corollary 3.
A paratopological group H is ♭ -separated if and only if H is a ( ♭ -closed)subgroup of a saturated paratopological group. (cid:3) Unfortunately we do not know the answer to the obvious ♭ -regular version of the abovecorollary. Problem 1.
Is every ♭ -regular paratopological group a subgroup of a regular saturatedparatopological group? For first-countable paratopological SIN-groups the answer to this problem is affirmative.
Corollary 4.
A first-countable paratopological SIN-group H is ♭ -regular if and only if H is a ♭ -closed subgroup of a regular first-countable saturated paratopological SIN-group G with | G/H | = ℵ .Proof. Taking into account that H is a first-countable paratopological SIN-group andapplying [BR , Proposition 3], we conclude that H ♭ is a first-countable topological SIN-group. Let T be the quotient group Q / Z of the group of rational numbers, endowed withthe Sorgenfrey topology generated by the base consisting of half-intervals. Observe that T is a ♭ -regular Sorgenfrey abelian paratopological group and the class of first countabletopological SIN-groups is T ♭ stable, closed-hereditary and contains H ♭ . Now to finish theproof apply Theorem 2. (cid:3) Next, we apply Theorems 1 and 2 to the class TBG (resp. FCTBG) of (first countable)totally bounded topological groups. We remind that a paratopological group G is totallybounded if for any neighborhood U of the unit in G there is a finite subset F ⊂ G with U F = F U = G . It is known that each totally bounded paratopological groupis saturated and a saturated paratopological group G is totally bounded if and only ifits group reflexion G ♭ is totally bounded, see [Ra ], [BR ]. An example of a ♭ -regulartotally bounded Sorgenfrey paratopological group is the circle T = { z ∈ C : | z | = 1 } endowed with the Sorgenfrey topology generated by the base consisting of half-intervals { z ∈ T : a ≤ Arg( z ) < b } where 0 ≤ a < b ≤ π . N SUBGROUPS OF SATURATED PARATOPOLOGICAL GROUPS 5
We define a paratopological group G to be Bohr separated (resp.
Bohr regular , fcBohrregular ) if it is TBG-separated (resp. TBG-regular, FCTBG-regular). In this terminologyTheorem 1 implies Corollary 5.
A paratopological group H is Bohr separated if and only if H is a ( ♭ -closed)subgroup of a totally bounded paratopological group. (cid:3) It is interesting to compare Corollary 5 with another characterizing theorem supply-ing us with many pathological examples of pseudocompact paratopological groups. Weremind that a topological space X is pseudocompact if each locally finite family of non-empty open subsets of X is finite. It should be mentioned that a Tychonoff space X ispseudocompact if and only if each continuous real-valued function on X is bounded. Theorem 3.
An abelian paratopological group H is Bohr separated if and only if H is asubgroup of a pseudocompact abelian paratopological group G with χ ( G ) = χ ( H ) . The following characterization of fcBohr regular paratopological groups follows fromTheorem 2 applied to the class G = FCTBG and the quotient group T = Q / Z endowedwith the standard Sorgenfrey topology. Corollary 6.
A paratopological group H is fcBohr regular if and only if H is a ♭ -closedsubgroup of a regular totally bounded paratopological group G with ℵ = χ ( G ♭ ) ≤ χ ( G ) = χ ( H ) and | G/H | ≤ ℵ . In some cases the fcBohr (= FCTBG) regularity is equivalent to the Bohr (= TBG)regularity. We recall that a topological space X has countable pseudocharacter if each onepoint subset of X is a G δ -subset. Proposition 2.
A Bohr regular paratopological group H is fcBohr regular provided oneof the following conditions is satisfied: (1) H is a k ω -space with countable pseudocharacter; (2) H is first countable and Lindel¨of.Proof. Using the Bohr regularity of H , find a regular bijective continuous homomorphism h : H → K onto a totally bounded topological group K . Denote by e H and e K the unitsof the groups H, K , respectively.1. First assume that H is a k ω -space with countable pseudo-character. In this casethe set H \ { e H } is σ -compact as well as its image h ( H \ { e H } ) = K \ { e K } . It followsthat the totally bounded group K has countable pseudo-character. Now it is standard tofind a bijective continuous homomorphism i : K → G of K onto a first countable totallybounded topological group G , see [Tk, 4.5]. Since the composition f ◦ h : H → G isbijective, the group H is FCTBG-separated and by Proposition 1 is fcBohr regular.2. Next assume that H is first-countable and Lindel¨of. Fix a sequence ( U n ) n ∈ ω ofopen subsets of H forming a neighborhood base at e H . For every n ∈ ω fix a closedneighborhood W n ⊂ U n whose image h ( W n ) is closed in K . Let us call an open subset U ⊂ K cylindrical if U = f − ( V ) for some continuous homomorphism f : K → G into afirst countable compact topological group G and some open set V ⊂ G . It follows fromthe total boundedness of K that open cylindrical subsets form a base of the topology ofthe group K , see [Tk, 3.4].Using the Lindel¨of property of the set f ( H \ U n ), for every n ∈ ω find a countablecover U n of f ( H \ U n ) by open cylindrical subsets such that ∪U n ∩ h ( W n ) = ∅ . Then TARAS BANAKH AND SASHA RAVSKY U = S n ∈ ω U n is a countable collection of open cylindrical subsets and we can produce acontinuous homomorphism f : K → G onto a first countable totally bounded topologicalgroup such that each set U ∈ U is the preimage U = f − ( V ) of some open set V ⊂ K .To finish the proof it rests to observe that the composition f ◦ h : H → G is a regularbijection of H onto a first countable totally bounded topological group. (cid:3) It can be shown that the character of any non-locally compact paratopological k ω -group with countable pseudo-character is equal to the small cardinal d , well-known inthe Modern Set Theory, see [JW], [Va]. By definition, d is equal to the cofinality of N ω endowed with the natural partial order: ( x i ) i ∈ ω ≤ ( y i ) i ∈ ω iff x i ≤ y i for all i . Moreprecisely, d is equal to the smallest size of a subset C ⊂ N ω cofinal in the sense that forany x ∈ N ω there is y ∈ C with y ≥ x . It is easy to see that ℵ ≤ d ≤ c . The MartinAxiom implies d = c . On the other hand, there are models of ZFC with d < c , see [Va].We shall apply Corollary 6 to construct examples of paratopological groups whosecharacter exceed their π -weight as well as the character of their group reflexions. Werecall that the π -weight πw ( X ) of a topological space X is the smallest size of a π -base ,i.e., a collection B of open nonempty subsets of X such that each nonempty open subset U of X contains an element of the family B . According to [Tk, 4.3] the π -weight of eachtopological group coincides with its weight. Corollary 7.
For any uncountable cardinal κ ≤ d there is a ♭ -regular totally boundedcountable abelian paratopological group G with ℵ = χ ( G ♭ ) = πw ( G ) < χ ( G ) = κ .Proof. Take any non-metrizable countable abelian FCTBG-separated topological k ω -group( H, τ ) (for example, let H be a free abelian group over a convergent sequence, see [FT]).The group H , being FCTBG-separated, admits a bijective continuous homomorphism h : H → K onto a first countable totally bounded abelian topological group K . ByProposition 1 this homomorphism is regular.According to [FT, 22)] or [Ban], the space H contains a copy of the Fr´echet-Urysohnfan S ω and thus has character χ ( H ) ≥ χ ( S ω ) ≥ d . Using this fact and the countabilityof H , by transfinite induction (over ordinals < κ ) construct a group topology σ ⊂ τ on H such that χ ( H, σ ) = κ and the homomorphism h : ( H, σ ) → K is regular. This meansthat the group ( H, σ ) is fcBohr regular. Now we can apply Corollary 6 to embed thegroup (
H, σ ) into a totally bounded countable abelian paratopological group G such that ℵ = χ ( G ♭ ) < χ ( G ) = χ ( H, σ ) = κ . Since G is saturated and abelian, ♭ -open subsets of G form a π -base which implies πw ( G ) = ω ( G ♭ ) = ℵ . (cid:3) Remark 1.
It is interesting to compare Corollary 7 with a result of [BRZ] asserting thatthere exists a ♭ -regular countable paratopological group G with ℵ = χ ( G ) < χ ( G ♭ ) = d .Such a paratopological group G cannot be saturated since χ ( G ♭ ) ≤ χ ( G ) for any saturated(more generally, any 2-oscillating) paratopological group G , see [BR ].We finish our discussion with presenting examples of regular (para)topological groupswhich embed into totally bounded paratopological groups but fail to embed into regular totally bounded paratopological groups. For that it suffices to find a Bohr separatedgroup which is not Bohr regular.Let us remark that each locally convex linear topological space (or more generallyeach locally quasi-convex abelian topological group, see [A] or [Ba]) is Bohr regular. Onthe other hand, there exist (non-locally convex) linear metric spaces which fail to beBohr separated or Bohr regular. The simplest example can be constructed as follows. N SUBGROUPS OF SATURATED PARATOPOLOGICAL GROUPS 7
Consider the linear space C [0 ,
1] of continuous real-valued functions on the closed interval[0 ,
1] and endow it with the invariant metrics d / ( f, g ) = R p | f ( t ) − g ( t ) | dt , p ( f, g ) = P n ∈ ω min { − n , | f ( t n ) − g ( t n ) |} and ρ ( f, g ) = d / ( f, g ) + p ( f, g ) where { t n : n ∈ ω } is an enumeration of rational numbers of [0 , C [0 , , d / ) admits no non-zero linear continuous functional and fails to be Bohrseparated. The linear metric space ( C [0 , , ρ ) is even more interesting. We remind thatan abelian group G is divisible (resp. torsion-free ) if for any a ∈ G and natural n theequation x n = a has a solution (resp. has at most one solution) x ∈ G . Proposition 3.
The linear metric space L = ( C [0 , , ρ ) is Bohr separated but fails tobe Bohr regular. Moreover, L is a ♭ -closed subgroup of a totally bounded abelian torsion-free divisible group, but fails to be a subgroup of a regular totally bounded paratopologicalgroup.Proof. The Bohr separatedness of L follows from the continuity of the maps χ n : L → R , χ n : f f ( t n ), for n ∈ ω . Let us show that the group L fails to be Bohr regular.For this we first prove that each linear continuous functional ψ : ( L, ρ ) → R iscontinuous with respect to the “product” metric p . Consider the open convex subset C = ψ − ( − ,
1) of L . By the continuity of ψ , there are n ≥ ε > x ∈ C for any x ∈ L with d / ( x, < ε and | x ( t i ) | < ε for all i ≤ n . Let L = { x ∈ L : x ( t i ) = 0for all i ≤ n } and observe that the convex set C ∩ L contains the open ε -ball with respectto the restriction of the metric d / on L . Now the standard argument (see [Ed, 4.16.3])yields C ∩ L = L and L ⊂ T k ≥ k C = ψ − (0). Hence the functional ψ factors throughthe quotient space L/L and is continuous with respect to the metric p (this follows fromthe continuity of the quotient homomorphism L → L/L with respect to p ).If χ : L → T is any character on L (that is a continuous group homomorphism into thecircle T = R / Z ), then it is easy to find a continuous linear functional ψ : L → R suchthat χ = π ◦ ψ , where π : R → T is the quotient homomorphism. As we have alreadyshown, the functional ψ is continuous with respect to the metric p and so is the character χ . Finally, we are able to prove that the group L fails to be Bohr regular. Assuming theconverse we would find a continuous regular homomorphism h : L → H onto a totallybounded abelian topological group H . The group H , being abelian and totally bounded,is a subgroup of the product T κ for some cardinal κ , see [Mo]. Then the above discussionyields that h is continuous with respect to the metric p . In this situation the regularity of h implies the regularity of the identity map ( L, ρ ) → ( L, p ). But this map certainly is notregular: for any 2 − n -ball B = { x ∈ L : ρ ( x, < − n } its closure in the metric p containsthe linear subspace { x ∈ L : x ( t i ) = 0 for all i ≤ n } and thus lies in no ball. Thereforethe group L is Bohr separated but not Bohr regular.Let G be the class of all totally bounded abelian divisible torsion-free topological groups.The group L , being Bohr separated, abelian, divisible, and torsion-free, is G -separated.Pick any irrational number α ∈ T = R / Z and consider the subgroup T = { qα : q ∈ Q } ofthe circle T endowed with the Sorgenfrey topology. It is clear that T is a totally bounded ♯ -discrete paratopological group with T ♭ ∈ G . By Theorem 1, L is a ♭ -closed subgroup of asaturated paratopological groups G with G ♭ ∈ G which implies that G is totally boundedabelian, divisible and torsion-free.On the other hand, L admits no embedding into a regular totally bounded paratopo-logical group G . Indeed, assuming that L ⊂ G is such an embedding, apply Theorem 3 TARAS BANAKH AND SASHA RAVSKY of [BR ] to conclude that the identity homomorphism id : G → G ♭ is regular and so is itsrestriction id | L , which would imply the Bohr regularity of L . (cid:3) There is also an alternative method of constructing Bohr separated but not Bohr reg-ular paratopological groups, based on the concept of a Lawson paratopological group.Following [BR ] we define a paratopological group G to be Lawson if it has a neighbor-hood base at the unit consisting of subsemigroups of G . According to [BR ] there is aregular Lawson paratopological group failing to be ♭ -separated. On the other hand, thereare Lawson paratopological groups which are ♭ -regular and Bohr separated but are nottopological groups, see Example 2 [BR ] or Example 1 below. We shall show that a ♭ -regular paratopological group G is a topological group provided its group reflexion G ♭ istopologically periodic. We remind that a paratopological group G is topologically periodic if for each x ∈ G and a neighborhood U ⊂ G of the unit there is a number n ≥ x n ∈ U , see [BG]. It is easy to show that each totally bounded topological group istopologically periodic. For paratopological groups it is not true: according to Theorem 2there is a ♭ -regular totally bounded paratopological group G which contains the discretegroup Z of integers and thus cannot be topologically periodic. The class of topologicallyperiodic topological groups will be denoted by TPTG. Proposition 4.
Each
TPTG -regular Lawson paratopological group is a topological group.Proof.
Let (
G, τ ) be a Lawson paratopological group and σ ⊂ τ be a topology turning G into a topologically periodic topological group such that ( G, τ ) has a base B at theunit consisting of subsemigroups, closed in the topology σ . We are going to show thatan arbitrary element U ∈ B is in fact a subgroup of G . For this purpose suppose thatthere exists an element x ∈ U − \ U . Then x − ∈ U and x m ∈ U for all m < U is a subsemigroup of G . Since the set U is closed in the topology σ , there exists aneighborhood V ∈ σ of unit such that xV ∩ U = ∅ . By the topological periodicity of( G, σ ), there exists a number n < − x n ⊂ V . Then x n +1 ∩ U = ∅ which is acontradiction. (cid:3) Since each totally bounded topological group is topologically periodic this Propositionimplies
Corollary 8.
Each Bohr regular Lawson paratopological group is a topological group. (cid:3)
On the other hand, abelian
Lawson paratopological groups are Bohr separated.
Proposition 5.
Each abelian Lawson paratopological group is Bohr separated.Proof.
Let G be such the group. Then G ♭ has a neighborhood base B at the unit, consistingof subgroups. For every group H ∈ B the group G/H , being abelian and discrete, is Bohrseparated [Mo]. Since the family { G → G/H : H ∈ B} of quotient maps separates thepoints of the group G , the group G is Bohr separated too. (cid:3) Corollary 8 and Proposition 5 allow us to construct simple examples of Bohr separatedLawson paratopological groups which are not Bohr regular.
Example 1.
There is a countable ♭ -regular saturated Lawson paratopological abelian group H which is Bohr separated but not Bohr regular. The group H has the following properties: (1) H is a ♭ -closed subgroup of a countable first-countable abelian totally boundedparatopological group; N SUBGROUPS OF SATURATED PARATOPOLOGICAL GROUPS 9 (2) H is a ♭ -closed subgroup of a first-countable abelian pseudocompact paratopologicalgroup; (3) H fails to be a subgroup of a regular totally bounded (or pseudocompact) paratopo-logical group.Proof. Consider the direct sum Z ω = { ( x i ) i ∈ ω ∈ Z ω : x i = 0 for all but finitely manyindices i } of countably many copies of the group Z of integers. Endow the group Z ω witha shift invariant topology τ whose neighborhood base at the origin consists of the sets U n = { } ∪ S m ≥ n W m where W m = { ( x i ) i ∈ ω ∈ Z ω : x i = 0 for all i < m and x m > } for m ≥
0. It is easy to see that H = ( Z ω , τ ) is a ♭ -regular countable first-countable saturatedLawson paratopological group which is not a topological group. By Proposition 5 andCorollary 8 the group H is Bohr separated but not Bohr regular.By Theorem 1, H is a ♭ -closed subgroup of a first-countable totally bounded countableparatopological group and by Theorem 3, H is a ♭ -closed subgroup of a first-countableabelian pseudocompact paratopological group.Assuming that H is a subgroup of a regular totally bounded or pseudocompact paratopo-logical group G and applying Theorem 3 of [BR ] and [RR] we would get that both G and H are Bohr regular which is impossible. (cid:3) In the proofs of our principal results we shall often exploit the following characterizationof semigroup topologies on groups from [Ra , 1.1]. Lemma 1.
A family B of subsets containing a unit e of a group G is a neighborhoodbase at e of some semigroup topology τ on G if and only if B satisfies the following fourPontryagin conditions:1. ( ∀ U, V ∈ B )( ∃ W ∈ B ) : W ⊂ U ∩ V ;2. ( ∀ U ∈ B )( ∃ V ∈ B ) : V ⊂ U ;3. ( ∀ U ∈ B )( ∀ x ∈ U )( ∃ V ∈ B ) : xV ⊂ U ;4. ( ∀ U ∈ B )( ∀ x ∈ G )( ∃ V ∈ B ) : x − V x ⊂ U .The topology τ is Hausdorff if and only if5. T { U U − : U ∈ B} = { e } . Proof of Theorem 1
The necessity is evident. We shall prove the sufficiency. Let (
H, τ ) be a G -separatedparatopological group, where G is T ♭ -stable class of topological groups. Since the group H is G -separated, there exists a group topology σ on the group H such that ( H, σ ) ∈ G .We shall define the topology on the product G = H × T as follows. Let B τ , B σ and B T be open bases at the unit of the groups ( H, τ ), (
H, σ ) and T respectively. For arbitraryneighborhoods U τ ∈ B τ , U σ ∈ B σ and U T ∈ B T with U τ ⊂ U σ put [ U τ , U σ , U T ] = U τ ×{ e T } ∪ U σ × ( U T \{ e T } ), where e H and e T are the units of the groups H and T respectively.The family of all such [ U τ , U σ , U T ] will be denoted by B . Now we verify the Pontryaginconditions for the family B .The Condition 1 is trivial.To check Condition 2 consider an arbitrary set [ U τ , U σ , U T ] ∈ B . There exist neighbor-hoods V τ ∈ B τ , V σ ∈ B σ such that V τ ⊂ U τ , V σ ⊂ U σ and V τ ⊂ V σ . Since the group T isdiscrete then there is a neighborhood V T ⊂ B T such that ( V T \{ e T } ) ⊂ U T \{ e T } . Then[ V τ , V σ , V T ] ⊂ [ U τ , U σ , U T ].To verify Condition 3 consider an arbitrary point x ∈ [ U τ , U σ , U T ] ∈ B . If x = ( x H , e T ),where x H ∈ U τ then there exist neighborhoods V τ ∈ B τ , V σ ∈ B σ such that V τ ⊂ V σ , x H V τ ⊂ U τ and x H V σ ⊂ U σ . Then x [ V τ , V σ , U T ] ⊂ [ U τ , U σ , U T ]. If x = ( x H , x T ), where x H ∈ U σ and x T ∈ U T \{ e T } then there exist neighborhoods V τ ∈ B τ , V σ ∈ B σ and V T ∈ B T such that V τ ⊂ V σ , x H V σ ⊂ U σ and x T V T ⊂ U T \{ e T } . Then x [ V τ , V σ , V T ] ⊂ [ U τ , U σ , U T ].Condition 4. Let x = ( x H , x T ) ⊂ H × T be an arbitrary point. Then there areneighborhoods V τ ∈ B τ , V σ ∈ B σ and V T ∈ B T such that V τ ⊂ V σ , x − H V τ x H ⊂ U τ , x − H V σ x H ⊂ U σ and x − T V T x T ⊂ U T . Then x − [ V τ , V σ , V T ] x ⊂ [ U τ , U σ , U T ].Hence the family B is a base of a semigroup topology on the group G . Denote thissemigroup topology by ρ . The inclusion T { [ U τ , U σ , U T ] · [ U τ , U σ , U T ] − : U τ ∈ B τ , U σ ∈B σ , U T ∈ B T } ⊂ { U σ U − σ × U T U − T : U σ ∈ B σ , U T ∈ B T } = { ( e H , e T ) } implies that thetopology ρ is Hausdorff. Since the groups T and ( H, σ ) are saturated and the group T isnondiscrete, the group ( G, ρ ) is saturated too. According to [BR , Proposition 3] the baseat the unit of the topology ρ ♭ consists of the sets U U − , where U ∈ B . Thus the topology ρ ♭ coincides with the product topology of the groups ( H, σ ) × T ♭ and hence ( G, ρ ♭ ) ∈ G and H is a ♭ -closed subgroup of the group G .2. Proof of Theorem 2
The “if” part of Theorem 2 is trivial. To prove the “only if” part, suppose that T and( H, τ ) are paratopological groups with the units e T and e H , satisfying the hypothesis ofTheorem 2.Using the Sorgenfrey property of the group T , choose an open invariant neighborhood U of the unit e T such that for any neighborhood U ⊂ T of e T there is a neighborhood U ′ ⊂ T of e T such that x, y ∈ U for any elements x, y ∈ U with xy ∈ U ′ . By inductionwe can build a sequence { U n : n ∈ ω } of invariant open neighborhoods of e T satisfyingthe following conditions:(1) { U n : n ∈ ω } is a neighborhood base at the unit e T of the group T ;(2) U n +1 ⊂ U n for every n ∈ ω ;(3) for every n ∈ ω and any points x, y ∈ U the inclusion xy ∈ U n +1 implies x, y ∈ U n ;(4) U n♭ $ U n − for every n ∈ ω , where U n♭ denotes the closure of the set U n in thetopology of T ♭ .Remark that the condition (3) yields(5) ( U \ U n ) U ∩ U n +1 = ∅ and hence U \ U n ∩ U n +1 U − = ∅ for all n .Since the group T is saturated, we can apply Proposition 3 of [BR ] to conclude that theset U n +2 U − is a neighborhood of the unit in T ♭ . Then the set U n +2 U n +2 U − ⊂ U n +1 U − is a neighborhood of U n +2 in T ♭ . This observation together with (5) yields(6) U \ U n♭ ∩ U n +2 = ∅ for all n .It follows from our assumptions on ( H, τ ) that there exists a group topology σ ⊂ τ on H such that the group ( H, σ ) belongs to the class G and ( H, τ ) has a neighborhood base B τ at the unit e H consisting of sets, closed in the topology σ . By induction we can builda base { V n : n ∈ ω } of open symmetric invariant neighborhoods of e H in the topology σ such that V n +1 ⊂ V n for every n ∈ ω .Consider the product H × T and identify H with the subgroup H × { e T } of H × T .It rests to define a topology on H × T . At first we shall introduce an auxiliary sequence { W k } of “neighborhoods” of ( e H , e T ) satisfying the Pontryagin Conditions 1,2, and 4. Forevery k ∈ ω let( ⋆ ) W n = { ( e H , e T ) } ∪ S i> n V ni × ( U i − \ U i ) N SUBGROUPS OF SATURATED PARATOPOLOGICAL GROUPS 11 and observe that W n +1 ⊂ W n for all n . Let us verify the Pontryagin Conditions 1,2,4 forthe sequence ( W n ).To verify Conditions 1 and 2 it suffices to show that W n ⊂ W n − for all n ≥
1. Fixany elements ( x, t ) , ( x ′ , t ′ ) ∈ W n . We have to show that ( xx ′ , tt ′ ) ∈ W n − . Without loss ofgenerality, we can assume that t, t ′ = e T . In this case we may find numbers i, i ′ > n with( x, t ) ∈ V ni × ( U i − \ U i ) and ( x ′ , t ′ ) ∈ V ni ′ × ( U i ′ − \ U i ′ ). For j = min { i, i ′ } the Conditions(2), (5) imply( xx ′ , tt ′ ) ∈ V nj − × ( U j − \ U j +1 ) ⊂ j +1 [ k = j − V ( n − k × ( U k − \ U k ) ⊂ [ k> n − V ( n − k × ( U k − \ U k ) ⊂ W n − . Taking into account that both the sequences { U n } and { V n } consist of invariant neigh-borhoods, we conclude that the sets W n are invariant as well. Hence the Condition 4holds too.Now, using the sequence ( W n ) we shall produce a sequence ( O n ) satisfying all thePontryagin Conditions 1–5. For every n ∈ ω put O n = S ∞ i = n W n W n +1 · · · W i . Thus W n ⊃ O n +1 ⊃ W n +1 and O n ∩ H × { e T } = { ( e H , e T ) } for all n . It is easy to see that the sequence { O n } consists of invariant sets and satisfies Pontryagin conditions 1–4. Hence the family { O n } is a neighborhood base at the unit of some (not necessarily Hausdorff) topology τ ′ on G = H × T turning G into a paratopological SIN-group. Applying Proposition 1.3from [Ra ] we conclude that the family B ρ = { OU : O ∈ B τ ′ , U ∈ B τ } is a neighborhoodbase at the unit of some (not necessarily Hausdorff) semigroup topology ρ on G (herewe identify H with the subgroup H × { e T } in G ). Since the topology ρ is stronger thanthe product topology π of the group ( H, σ ) × T ♭ , the topology ρ is Hausdorff and H is a ♭ -closed subgroup of the group ( G, ρ ). It follows from the construction of the topology ρ that ρ | H = τ , χ ( G, ρ ) = χ ( H ) and | G/H | = | T | .At the end of the proof we show that the paratopological group ( G, ρ ) is saturated and ♭ -regular. To show that the group ( G, ρ ) is saturated it suffices to find for every n ≥ V ⊂ ( H, σ ) and U ⊂ T such that V × U − ⊂ W n . Taking into accountthat the group T is saturated and the set U n − \ U n♭ is nonempty, find a nonempty openset U ⊂ T such that U − ⊂ U n − \ U n♭ . Then V − n × U − ⊂ V n × ( U n − \ U n ) ⊂ W n .This implies that the group ( G, ρ ) is saturated and (
G, ρ ♭ ) = ( H, σ ) × T ♭ ∈ G .The ♭ -regularity of the group ( G, ρ ) will follow as soon as we prove that W n V π ⊂ W n − V for every n ≥ V ∈ B τ . Indeed, in this case, we shall get O n +1 V ♭ ⊂ O n +1 V π ⊂ W n V π ⊂ W n − V ⊂ O n − V. Fix any x ∈ W n V π . If x ∈ V ×{ e T } , then x ∈ W n − V . Next, assume that x / ∈ H ×{ e T } .The property (4) of the sequence ( U k ) implies that the point x has a π -neighborhoodmeeting only finitely many sets H × U i , i ∈ ω . This observation together with x ∈ W n V π and ( ⋆ ) imply that x ∈ V ni V × ( U i − \ U i ) ♭ for some i > n . The condition (6) implies that the following chain of inclusions holds: x ∈ V ni V × ( U i − \ U i ) ♭ ⊂ V ni V σ × ( U i − \ U i ) ♭ ⊂ V ni V × ( U i − \ U i +2 ) ⊂ i +2 [ j = i − V ni − V × ( U j − \ U j ) ⊂ [ j> n − V ( n − j V × ( U j − \ U j ) ⊂ W n − V. Finally, assume that x ∈ H \ V = ( H \ V ) × { e T } . Since the set V is ♭ -closed in H ,there is m ∈ ω such that V − m V m x ∩ V = ∅ and thus V m x ∩ V i V = ∅ for all i ≥ m . Theinclusion x ∈ W n V π and ( ⋆ ) imply( V m × U m U − m ) x ∩ ( V ni V × ( U i − \ U i )) = ∅ for some i > n . Then V m x ∩ V ni V = ∅ and U m U − m ∩ ( U i − \ U i ) = ∅ . In view of Property(5) of the sequence ( U k ), the latter relation implies m ≤ i . On the other hand, the formerrelation together with the choice of the number m yields ni < m ≤ i which is impossible.This contradiction finishes the proof of the inclusion W n V π ⊂ W n − V .3. Proof of Theorem 3
Given a topological space (
X, τ ) Stone [Sto] and Katetov [Kat] considered the topology τ r on X generated by the base consisting of all canonically open sets of the space ( X, τ ).This topology is called the regularization of the topology τ . If ( X, τ ) is Hausdorff then(
X, τ r ) is regular and if ( X, τ ) is a paratopological group then (
X, τ r ) is a paratopologicalgroup too [Ra , Ex.1.9]. If ( G, τ ) is a paratopological group then τ r is the strongestregular semigroup topology on the group G which is weaker than τ ; moreover, for anyneighborhood base B at the unit of the group ( G, τ ) the family B r = { int U : U ∈ B} isa base at the unit of the group ( G, τ r ) [Ra , p.31–32]. The following proposition is quiteeasy and probably is known. Proposition 6.
Let ( X, τ ) be a topological space. Then ( X, τ ) is pseudocompact if andonly if the regularization ( X, τ r ) is pseudocompact. For the proof of Theorem 3 we shall need a special pseudocompact functionally Haus-dorff semigroup topology on the unit circle. We recall that a topological space X is functionally Hausdorff if continuous functions separate points of X . Proposition 7.
There is a functionally Hausdorff pseudocompact first countable semi-group topology θ on the unit circle T which is not a group topology.Proof. Let T be the unit circle and χ : T → Q be a (discontinuous) group homomorphismonto the groups of rational numbers. Fix any element x ∈ T with χ ( x ) = 1 and observethat S = { }∪{ x ∈ T : χ ( x ) > } is a subsemigroup of T . Let θ be the weakest semigrouptopology on T containing the standard compact topology τ and such that S is open in θ .It is easy to see that θ is functionally Hausdorff and the sets S ∩ U , where 1 ∈ U ∈ τ ,form a neighborhood base of the topology θ at the unit of T .By Proposition 6, to show that the group ( T , θ ) is pseudocompact it suffices to verifythat θ r = τ . Since τ is a regular semigroup topology on the group T weaker than θ ,we get θ r ⊃ τ . To verify the inverse inclusion we first show that U τ = U θ for any U ∈ θ . Since τ ⊂ θ it suffices to show that U τ ⊂ U θ . Fix any point x ∈ U τ and aneighborhood V ∈ τ of 1. We have to show that x ( V ∩ S ) ∩ U = ∅ . Pick up any point y ∈ xV ∩ U . Since U is open in the topology θ , we can find a neighborhood W ∈ τ of N SUBGROUPS OF SATURATED PARATOPOLOGICAL GROUPS 13 y ( W ∩ S ) ⊂ xV ∩ U . Find a number N such that χ ( yx N ) > χ ( x ) and thus yx n ∈ xS for all n ≥ N (we recall that x is an element of T with χ ( x ) = 1). Moreover,since x is non-periodic in T , there exists a number n ≥ N such that x n ⊂ W . Then yx n ∈ ( yS ∩ yW ) ∩ xS ⊂ ( xV ∩ U ) ∩ xS = x ( V ∩ S ) ∩ U . Hence x ∈ U θ and U θ = U τ .Then int θ U θ = T \ T \ U θθ = T \ T \ U θτ ∈ τ which just yields θ r ⊂ τ . (cid:3) Now we are able to present a proof of Theorem 3 . The “if” part follows from the ob-servation that for any Hausdorff pseudocompact paratopological group (
G, τ ) its groupreflexion G ♭ = ( G, τ r ) is a Hausdorff pseudocompact (and hence totally bounded) topo-logical group [RR].To prove the “only if” part, fix a Bohr-separated abelian paratopological group ( H, τ )and let B τ be a neighborhood base at the unit of the group ( H, τ ). It follows that thereis a group topology σ ′ ⊂ τ on H such that ( H, σ ′ ) is totally bounded. Let ( ˆ H, σ ) be theRaikov completion of the group (
H, σ ′ ). It is clear that ˆ H is a compact abelian group and H is a normal dense subgroup of ˆ H . It follows that B τ is a neighborhood base at the unitof some semigroup topology τ ′ on the group ˆ H with τ ′ | H = τ . Let ( T , θ ) be the groupfrom Proposition 7.We shall define the topology on the product G = ˆ H × T as follows. Let B τ , B σ and B θ be the open neighborhood bases at the unit of the groups ( H, τ ), ( ˆ
H, σ ) and( T , θ ) respectively. For arbitrary neighborhoods U τ ∈ B τ , U σ ∈ B σ and U θ ∈ B θ with U τ ⊂ U σ let [ U τ , U σ , U θ ] = U τ × { e T } ∪ U σ × ( U θ \{ e T } ), where e H and e T are the unitsof the groups H and T respectively. Denote by B the family of all such [ U τ , U σ , U θ ].Repeating the argument of Theorem 1 check that the family B is a base of some Hausdorffsemigroup topology ρ on G . By π denote the topology of the product ( ˆ H, σ ) × ( T , θ r ).By Proposition 6 to show that the group ( G, ρ ) is pseudocompact it suffice to verify that ρ r ⊂ π . For this we shall show that U ρ ⊃ U σ × U θθ for every U = [ U τ , U σ , U θ ] ∈ B .Let ( x ˆ H , x T ) ∈ U σ × U θθ and V = [ V τ , V σ , V θ ] ∈ B . It suffice to show that (cid:0) ( x ˆ H , x T ) + V σ × ( V θ \{ e T } ) (cid:1) ∩ U σ × ( U θ \{ e T } ) = ∅ . This intersection is nonempty if and only if theintersections ( x ˆ H + V σ ) ∩ U σ and ( x T + ( V θ \{ e T } )) ∩ ( U θ \{ e T } ) are nonempty. The firstintersection is nonempty since x ˆ H ∈ U σ and the second is nonempty since x T ∈ U θθ andthe topology θ is non-discrete. References [AH] A.V. Arhangel’skii, M. Huˇsek, Extensions of topological and semitopological groups and the productoperation //
Comment. Math. Univ. Carolinae. :1 (2001) 173-186.[A] L. Außenhofer, Contributions to the duality theory of abelian topological groups and to the theoryof nuclear groups // Dissert. Math. (1999).[Ban] T. Banakh, On topological groups containing a Fr´echet-Urysohn fan //
Matem. Studii . :2 (1998),149–154.[Ba] W. Banaszczyk, Additive Subgroups of Topological Vector Spaces , Lect. Notes in Math., Vol.1466,Springer, Berlin, 1991.[BR ] T. Banakh, O. Ravsky. Oscillator topologies on a paratopological group and related number in-variants, // Algebraical Structures and their Applications , Kyiv: Inst. Mat. NANU, (2002) 140-153.[BR ] T. Banakh, O. Ravsky. Asymmetry indices of paratopological groups, preprint.[BR ] T. Banakh, O. Ravsky. On completeness properties of paratopological groups, (in preparation).[BRZ] T. Banakh, S.Ravsky, L. Zdomsky. On the character of the group topology generated by a count-able T -filter, (in preparation). [BG] B. Bokalo, I. Guran. Sequentially compact Hausdorff cancellative semigroup is a topological group// Matem. Studii (1996), 39–40.[Cho] M.M. Choban. Some topics in topological algebra // Topology Appl. (1993), 183–202.[Ed] R.E.Edwards. Functional Analysis. Theory and Applications , NY: Holt, Rinehart and Winston,1965.[FT] S.P. Franklin, B.V. Smith Thomas. A survey of k ω -spaces // Topology Proc. (1977), 111–124.[Gra] M.I. Graev. Free topological groups // Izvestia AN SSSR. Ser. Mat. :3 (1948), 279–324.[Gu] I.Y. Guran. Cardinal invariants of paratopological groups // in: II Intern. Algebraic Conf. inUkraine , – Kyiv - Vinnytsia, 1999 (in Ukrainian).[JW] W. Just, M. Weese.
Discovering Modern Set Theory. II (Graduate Studies in Math. Vol. 18, Prov-idence: AMS, 1997.[Kat] M. Kat˘etov. On H-closed extensions of topological spaces // ˘Casopis P˘est. Mat. Fys. (1947),17–32.[Mo] S. A. Morris. Pontryagin duality and the structure of locally compact Abelian groups.
London Math-ematical Society Lecture Note Series 29. Cambridge University Press, 1977.[Ra ] O. Ravsky. Paratopological groups I // Matem. Studii. :1 (2001), 37–48.[Ra ] O. Ravsky. Paratopological groups II // Matem. Studii. :1 (2002), 93–101.[Ra ] O. Ravsky. The topological and algebraical properties of paratopological groups . Ph.D. Thesis. – LvivUniversity, 2003 (in Ukrainian).[Ra ] O.Ravsky, On H-closed paratopological groups // Visnyk Lviv Univ., Ser. Mekh.-Mat. (2003),172–179.[RR] O. Ravsky, E. Reznichenko. The continuity of inverse in groups // in: Intern. Conf. on Funct.Analysis and its Appl. Dedicated to the 110th anniversary of Stefan Banach (May 28-31, 2002) – Lviv,2002. – P.170–172.[Sto] M.H. Stone. Applications of the theory of Boolean rings to general topology // Trans. Amer. Math.Soc. (1937), 375–481.[Tk] M. Tkachenko. Introduction to topological groups // Topology Appl . (1998), 179–231.[Va] J.E. Vaughan. Small uncountable cardinals and topology // in: Open Problems in Topology (J. vanMill and G.M.Reed eds.), Amsterdam: North-Holland, 1990. – P.195–216.
E-mail address : [email protected] Instytut Matematyki, Akademia ´Swie¸tokrzyska in Kielce, ´Swie¸tokrzyska 15, Kielce,25406, PolandandDepartment of Mathematics, Ivan Franko Lviv National University, Universytetska, 1Lviv 79000, Ukraine.
E-mail address : [email protected]@mail.ru