Submaximal properties in (strongly) topological gyrogroups
aa r X i v : . [ m a t h . GN ] N ov SUBMAXIMAL PROPERTIES IN (STRONGLY) TOPOLOGICALGYROGROUPS
MENG BAO AND FUCAI LIN*
Abstract.
A space X is submaximal if any dense subset of X is open. In this paper,we prove that every submaximal topological gyrogroup of non-measurable cardinalityis strongly σ -discrete. Moreover, we prove that every submaximal strongly topologicalgyrogroup of non-measurable cardinality is hereditarily paracompact. Introduction
In the 1940s, E. Hewitt in [14] introduced the concepts of maximality and submax-imality of general topological spaces, which are important tools to deal with extremecases when studying the family of topologies without isolated points. At the same time,E. Hewitt found a general way to construct maximal and submaximal topologies. Heconstructed maximal and submaximal topologies by transfinite induction and by themethod that any chain of topologies on the same set has an upper bound with the sameseparation axioms. Then, A.V. Arhangel’ skiˇı and P.J. Collins [2] began to study theclass of submaximal spaces systematically in 1995 and give some necessary and sufficientconditions for a space to be submaximal. In 1998, O. Alas, I. Protasov, M. Tkachenko,etc.[3] studied the maximal and submaximal groups and proved that every submaxi-mal topological group of non-measurable cardinality is strongly σ -discrete, and everysubmaximal strongly topological group of non-measurable cardinality is hereditarilyparacompact.The gyrogroup was first introduced by A.A. Ungar in [21] when he researched c -ballof relativistically admissible velocities with Einstein velocity addition. The Einsteinvelocity addition ⊕ E is given as follows: u ⊕ E v = 11 + u · v c ( u + 1 γ u v + 1 c γ u γ u ( u · v ) u ) , where u , v ∈ R c = { v ∈ R : || v || < c } and γ u is given by γ u = 1 p − u · u c . From [23], we know that the gyrogroup has a weaker algebraic structure than a group.In 2017, W. Atiponrat [4] gave the concept of topological gyrogroups. Then Z. Cai, S.Lin and W. He in [8] proved that every topological gyrogroup is a rectifiable space. In2019, the authors [5] defined the concept of strongly topological gyrogroups, and proved
Mathematics Subject Classification.
Primary 54A20; secondary 11B05; 26A03; 40A05; 40A30;40A99.
Key words and phrases. topological gyrogroups; strongly topological gyrogroups; Submaximal prop-erties; paracompact.The authors are supported by the Key Program of the Natural Science Foundation of Fujian Province(No: 2020J02043), the NSFC (No. 11571158), the lab of Granular Computing, the Institute of Meteo-rological Big Data-Digital Fujian and Fujian Key Laboratory of Data Science and Statistics.*corresponding author. that every feathered strongly topological gyrogroup is paracompact. Moreover, theauthors proved that every strongly topological gyrogroup with a countable pseudochar-acter is submetrizable and every locally paracompact strongly topological gyrogroup isparacompact, see [6, 7].In this paper, we generalize some well known results in the class of submaximal topo-logical groups to submaximal topological gyrogroups. In particular, we prove that everysubmaximal topological gyrogroup of non-measurable cardinality is strongly σ -discreteand every submaximal strongly topological gyrogroup of non-measurable cardinality ishereditarily paracompact, which generalizes some results in [3].2. Preliminaries
Throughout this paper, all topological spaces are assumed to be Hausdorff and densein themselves, unless otherwise is explicitly stated. Let N be the set of all positiveintegers and ω the first infinite ordinal. Let X be a topological space and A ⊆ X be asubset of X . The closure of A in X is denoted by A and the interior of A in X is denotedby Int( A ). A cardinal number m is called non-measurable [10] provided that the onlycountably additive two-valued measure defined on the family of all subsets of a set X of cardinality m which vanishes on all one-point sets is the trivial measure, identicallyequal to zero. The readers may consult [1, 10, 17] for notation and terminology notexplicitly given here. Definition 2.1. [4] Let G be a nonempty set, and let ⊕ : G × G → G be a binaryoperation on G . Then the pair ( G, ⊕ ) is called a groupoid . A function f from a groupoid( G , ⊕ ) to a groupoid ( G , ⊕ ) is called a groupoid homomorphism if f ( x ⊕ y ) = f ( x ) ⊕ f ( y ) for all elements x, y ∈ G . Furthermore, a bijective groupoid homomorphism froma groupoid ( G, ⊕ ) to itself will be called a groupoid automorphism . We write Aut( G, ⊕ )for the set of all automorphisms of a groupoid ( G, ⊕ ). Definition 2.2. [22] Let ( G, ⊕ ) be a groupoid. The system ( G, ⊕ ) is called a gyrogroup ,if its binary operation satisfies the following conditions:(G1) There exists a unique identity element 0 ∈ G such that 0 ⊕ a = a = a ⊕ a ∈ G .(G2) For each x ∈ G , there exists a unique inverse element ⊖ x ∈ G such that ⊖ x ⊕ x = 0 = x ⊕ ( ⊖ x ).(G3) For all x, y ∈ G , there exists gyr[ x, y ] ∈ Aut( G, ⊕ ) with the property that x ⊕ ( y ⊕ z ) = ( x ⊕ y ) ⊕ gyr[ x, y ]( z ) for all z ∈ G .(G4) For any x, y ∈ G , gyr[ x ⊕ y, y ] = gyr[ x, y ]. Lemma 2.3. [22]
Let ( G, ⊕ ) be a gyrogroup. Then for any x, y, z ∈ G , we obtain thefollowing: (1) ( ⊖ x ) ⊕ ( x ⊕ y ) = y . (left cancellation law) (2) ( x ⊕ ( ⊖ y )) ⊕ gyr [ x, ⊖ y ]( y ) = x . (right cancellation law) (3) ( x ⊕ gyr [ x, y ]( ⊖ y )) ⊕ y = x . (4) gyr [ x, y ]( z ) = ⊖ ( x ⊕ y ) ⊕ ( x ⊕ ( y ⊕ z )) . The definition of a subgyrogroup is given as follows.
Definition 2.4. [19] Let ( G, ⊕ ) be a gyrogroup. A nonempty subset H of G is called a subgyrogroup , denoted by H ≤ G , if H forms a gyrogroup under the operation inheritedfrom G and the restriction of gyr [ a, b ] to H is an automorphism of H for all a, b ∈ H . UBMAXIMAL PROPERTIES IN (STRONGLY) TOPOLOGICAL GYROGROUPS 3
Furthermore, a subgyrogroup H of G is said to be an L -subgyrogroup , denoted by H ≤ L G , if gyr [ a, h ]( H ) = H for all a ∈ G and h ∈ H .The subgyrogroup criterion is given in [19] (that is, a nonempty subset H of a gy-rogroup G is a subgyrogroup if and only if ⊖ a ∈ H and a ⊕ b ∈ H for all a, b ∈ H )which explains that by the item (4) in Lemma 2.3 it follows the subgyrogroup criterion. Definition 2.5. [4] A triple (
G, τ, ⊕ ) is called a topological gyrogroup if the followingstatements hold:(1) ( G, τ ) is a topological space.(2) ( G, ⊕ ) is a gyrogroup.(3) The binary operation ⊕ : G × G → G is jointly continuous while G × G is endowedwith the product topology, and the operation of taking the inverse ⊖ ( · ) : G → G , i.e. x → ⊖ x , is also continuous.It is clear that each topological group is a topological gyrogroup. However, everytopological gyrogroup whose gyrations are not identically equal to the identity is not atopological group. Example 2.6. [4]
The Einstein gyrogroup with the standard topology is a topologicalgyrogroup but not a topological group.
The Einstein gyrogroup has been introduced in the Introduction. It was proved in[22] that ( R c , ⊕ E ) is a gyrogroup but not a group. Moreover, with the standard topol-ogy inherited from R , it is clear that ⊕ E is continuous. Finally, − u is the inverse of u ∈ R and the operation of taking the inverse is also continuous. Therefore, the Ein-stein gyrogroup ( R c , ⊕ E ) with the standard topology inherited from R is a topologicalgyrogroup but not a topological group. Definition 2.7. [14] A topological space (
X, τ ) is called maximal if for any topology µ on X strictly finer that τ , the space ( X, µ ) has an isolated point. A space X is submaximal if any dense subset of X is open. Definition 2.8. [3] A non-empty family D of dense subsets of a space X is called a filter of dense subsets of X if D is closed with respect to finite intersections and D ∈ D , D ⊂ D ⊂ X implies D ∈ D . The family D is called an ultrafilter of dense subsets of X if there is no filter of dense subsets of X that properly contains D . Definition 2.9. [3] A space is called irresolvable if it is not the union of two disjointdense subsets.
Definition 2.10. [10] A space X is called collectionwise Hausdorff if for any discretesubset A of X it is possible to choose an open set V p containing p for every p ∈ A insuch a way that the family { V p : p ∈ A } is discrete. Definition 2.11. [10] If X is a space and x ∈ X , then the dispersion character ∆( x, X )of X at the point x is the minimum of the cardinalities of open subsets of X containing x .The cardinal number ∆( X ) = min { ∆( x, X ) : x ∈ X } is called the dispersion character of X .Next, we recall the definition of strongly topological gyrogroups. Definition 2.12. [5] Let G be a topological gyrogroup. We say that G is a stronglytopological gyrogroup if there exists a neighborhood base U of 0 such that, for every U ∈ U , gyr[ x, y ]( U ) = U for any x, y ∈ G . For convenience, we say that G is a stronglytopological gyrogroup with neighborhood base U of 0. MENG BAO AND FUCAI LIN*
For each U ∈ U , we can set V = U ∪ ( ⊖ U ). Then, gyr [ x, y ]( V ) = gyr [ x, y ]( U ∪ ( ⊖ U )) = gyr [ x, y ]( U ) ∪ ( ⊖ gyr [ x, y ]( U )) = U ∪ ( ⊖ U ) = V, for all x, y ∈ G . Obviously, the family { U ∪ ( ⊖ U ) : U ∈ U } is also a neighborhood baseof 0. Therefore, we may assume that U is symmetric for each U ∈ U in Definition 2.12.In [5], the authors proved that there is a strongly topological gyrogroup which is nota topological group, see Example 2.13. Example 2.13. [5]
Let D be the complex open unit disk { z ∈ C : | z | < } . Weconsider D with the standard topology. In [4, Example 2] , define a M¨obius addition ⊕ M : D × D → D to be a function such that a ⊕ M b = a + b ab for all a, b ∈ D . Then ( D , ⊕ M ) is a gyrogroup, and it follows from [4, Example 2] that gyr [ a, b ]( c ) = 1 + a ¯ b ab c for any a, b, c ∈ D . For any n ∈ N , let U n = { x ∈ D : | x | ≤ n } . Then, U = { U n : n ∈ N } is aneighborhood base of . Moreover, we observe that | a ¯ b ab | = 1 . Therefore, we obtain that gyr [ x, y ]( U ) ⊂ U , for any x, y ∈ D and each U ∈ U , then it follows that gyr [ x, y ]( U ) = U by [19, Proposition 2.6] . Hence, ( D , ⊕ M ) is a strongly topological gyrogroup. However, ( D , ⊕ M ) is not a group [4, Example 2] . Indeed, it is well known that M¨obius gyrogroups, Einstein gyrogroups, and Propervelocity gyrogroups, that were studied in [11, 12, 13, 22], are all strongly topologicalgyrogroups. Therefore, they are all topological gyrogroups and rectifiable spaces. Atthe same time, all of them are not topological groups. Further, it was also proved in [5,Example 3.2] that there exists a strongly topological gyrogroup which has an infinite L -subgyrogroup.3. submaximal properties of topological gyrogroups In this section, we mainly prove that every submaximal topological gyrogroup of non-measurable cardinality is strongly σ -discrete. First, we show that, for any cardinality κ > ω , there exists a gyrogroup G with subgyrogroup H of the cardinality κ such that H is not a group. Example 3.1.
For any cardinality κ > ω , there exists a gyrogroup G with subgyrogroup H of the cardinality κ such that H is not a group. Let D be gyrogroup in Example 2.13 and let κ be an infinite cardinal number. Itfollows from [20, Theorem 2.1] that D κ is a gyrogroup. Fix a subset X of the gyrogroup D κ such that the cardinality of X is equal to κ and X contains arbitrary three points x = ( x α ) α<κ , y = ( y α ) α<κ and z = ( z α ) α<κ of D κ such that there exists β < α with x β = 1 / , y β = i/ z β = − /
2. From the proof of [4, Example 2], we see that x ⊕ ( y ⊕ z ) = ( x ⊕ y ) ⊕ z . Put H = h X i , that is, H is a subgyrogroup generated from X . Then the cardinality of H is also equal to κ . Moreover, since x, y, z ∈ H , it followsthat H is not a group. Proposition 3.2.
Let G be a gyrogroup of cardinality κ > ω . Then, for any ω < α < κ ,there exists a subgyrogroup G α of G with the cardinality α . UBMAXIMAL PROPERTIES IN (STRONGLY) TOPOLOGICAL GYROGROUPS 5
Proof.
Take an arbitrary subset Y of G such that Y = ⊖ Y , 0 ∈ Y and | Y | = α .Let Y = Y . By induction, we assume that we have defined Y , . . . , Y n of subsets of G such that Y i +1 = ⊖ ( Y i ⊕ Y i ) ∪ ( Y i ⊕ Y i ) for any i = 0 , . . . , n −
1. Let Y n +1 = ⊖ ( Y n ⊕ Y n ) ∪ ( Y n ⊕ Y n ). Clearly, the cardinality of each Y n is just α . Put G α = S n ∈ N Y n .Then G α is a subgyrogroup of G with the cardinality α . Indeed, it is obvious that | G α | = α . It suffices to prove that G α is a subgyrogroup of G . By our constructionof G α , we have G α = ⊖ G α . Take any x, y ∈ G α . Then there exists n ∈ N such that x, y ∈ Y n , hence x ⊕ y ∈ Y n +1 ⊂ G α . Therefore, G α is a subgyrogroup of G . (cid:3) Next we give some lemmas.
Lemma 3.3.
Let G be a gyrogroup of cardinality κ > ω . Then for each α < κ there aresubsets G α and H α of G with the following properties:(1) G α is a subgyrogroup of G for all α < κ ;(2) if α < β < κ , we have G α ⊂ G β and G α = G β ;(3) | G α | = | α | for all α < κ ;(4) G α = S { H υ : υ ≤ α } for all α < κ ;(5) S { H α : α < κ } = G and H α ∩ H β = ∅ if α = β ;(6) if g ∈ H α and α < β , we have that g ⊕ H β = H β ⊕ g = H β ;(7) H α = ⊖ H α for all α < κ ;(8) if A is a confinal subset of κ , the cardinality of S { H α : α ∈ A } is κ .The family { H α : α < κ } is called a canonical decomposition of G .Proof. Since G is a gyrogroup of cardinality κ , let G = { g α : α < κ } , where g = 0 and g α = g β if α = β . Let G = h{ g }i . Suppose that β < κ and that for every α < β wehave constructed a subgyrogroup G α of G which has the following properties:(i) G α ⊂ G γ and G α = G γ if α < γ < β ;(ii) | G α | = | α | for all α < β ;(iii) { g γ : γ < α } ⊂ G α for every α < β .Let B β = S { G α : α < β } . It follows from (ii) that B β = G , hence there exists β ∗ = min { α < κ : g α B β } . Set G β = h B β ∪ { g β ∗ }i . Therefore, by induction, we can obtain that the family { G α : α < κ } satisfies (i)-(iii) as well as the property S { G α : α < κ } . For every α < κ , let H α = G α \ S { G β : β < α } .Obviously, the sets G α and H α satisfy (1)-(5) and (7). To see that (6) holds. Assumethat g ∈ H α and α < β . Clearly, g ∈ G α and G α is a subgyrogroup of G γ for each α ≤ γ ≤ β . Therefore, g ⊕ G γ = G γ ⊕ g = G γ for all γ , then g ⊕ H β = g ⊕ ( G β \ [ { G α : α < β } )= ( g ⊕ G β ) \ [ { g ⊕ G α : α < β } = G β \ [ { G α : α < β } = H β . Similar, H β ⊕ g = H β , thus (6) holds. Finally, take any cofinal A ⊂ κ . It follows from | H α +1 | = | G α | = | α | that | [ { H α : α ∈ A }| = | [ { G α : α ∈ A }| = | G | = κ. MENG BAO AND FUCAI LIN* (cid:3)
Let G be a gyrogroup, and let τ be a topology on G . A left topological gyrogroup consists of a gyrogroup G and a topology τ on the set G such that for all g ∈ G ,the left action l g : G → G , x g ⊕ x , is a continuous mapping of the space G toitself. Similarly, we can define the concept of right topological gyrogroups . Clearly, eachtopological gyrogroup is not only a left topological gyrogroup but also a right topologicalgyrogroup.Let G be a gyrogroup of cardinality κ > ω , and let { H α : α < κ } be a canonicaldecomposition of G . Then for each A ⊂ κ put H A = S { H α : α ∈ A } . Lemma 3.4.
Suppose that ( G, τ, ⊕ ) is a non-discrete irresolvable left (or right) topo-logical gyrogroup such that | G | = ∆( G, τ, ⊕ ) = κ > ω , and suppose that { H α : α < κ } isa canonical decomposition of G . Then the following statements hold:(1) For each subset A ⊂ κ and any g, h ∈ G , the set ( h ⊕ ( g ⊕ H A )) \ H A hascardinality less than κ .(2) The family ξ = { A ⊂ κ : Int ( H A ) = ∅} is a free ultrafilter on κ .Proof. (1) Since g, h ∈ G , there exists α, β < κ such that g ∈ H α ⊂ G α = [ { H υ : υ ≤ α } and h ∈ H β ⊂ G β = S { H υ : υ ≤ β } . If α < γ ( β < γ ), it follows from Lemma 3.3 that g ⊕ H γ = H γ ( h ⊕ H γ = H γ ). If α ≥ β , then g ⊕ H A ⊂ G α ∪ { H ν : ν > α, ν ∈ A } . Hence h ⊕ ( g ⊕ H A ) ⊂ G α ∪ { H ν : ν > α, ν ∈ A } , then ( h ⊕ ( g ⊕ H A )) \ H A ⊂ G α . If α < β , then it also easily see that h ⊕ ( g ⊕ H A ) ⊂ G β ∪ { H ν : ν > β, ν ∈ A } . Hence ( h ⊕ ( g ⊕ H A )) \ H A ⊂ G β . Therefore, it follows from Lemma 3.3 that( h ⊕ ( g ⊕ H A )) \ H A has cardinality less than κ . (2) It is clear that H A and H κ \ A are disjoint and G = H A ∪ H κ \ A . Hence one ofthe sets H A or H κ \ A has non-empty interior as G is irresolvable. Therefore, A ∈ ξ or κ \ A ∈ ξ . Indeed, exactly one of the sets A and κ \ A belongs to ξ . Suppose not, thenboth A and κ \ A belong to ξ . In order to obtain a contradiction, it suffices to provethat H A and H κ \ A are dense in G . Indeed, we need only to consider the case of H A .Clearly, U = Int( H A ) = ∅ . If U is not dense in G , there exists a non-empty open set V ⊂ G such that V ∩ U = ∅ . For arbitrary x ∈ U and y ∈ V , set W = ( y ⊕ (( ⊖ x ) ⊕ U )) ∩ V .Obviously, W is open, non-empty and | W | = κ .By (1), we see that | W \ H A | < κ , hence Int( W \ H A ) = ∅ in G by our assumption.By the definitions of W and U , it follows that W = ( W \ H A ) ∪ ( W ∩ H A ) ⊂ ( W \ H A ) ∪ ( H A \ U ) , where W \ H A and H A \ U are disjoint and both of them have empty interior and densein W . Therefore, W is resolvable. The gyrogroup G can be covered by the all possibleleft translations of W , so G is resolvable by [9] which proves that the union of resolvablespaces is also resolvable. This is a contradiction. UBMAXIMAL PROPERTIES IN (STRONGLY) TOPOLOGICAL GYROGROUPS 7
Therefore, if U = Int( H A ) = ∅ , then it follows that U has to be dense in G . So A or κ \ A belong to ξ and ξ is an ultrafilter on κ . Further, ξ is a free ultrafilter since each H α does not belongs to ξ by (5) of Lemma 3.3. (cid:3) Theorem 3.5.
Suppose that ( G, τ, ⊕ ) is a non-discrete irresolvable left (or right) topo-logical gyrogroup of non-measurable cardinality such that ∆( G, τ, ⊕ ) = κ . If { H α : α <κ } is a canonical decomposition of G , then there is a family { A n : n ∈ N } of subsets of κ such that:(1) S { A n : n ∈ N } = κ ;(2) every set H A n = S { H α : α ∈ A n } is closed and nowhere dense in G ;(3) S { H A n : n ∈ N } = G .In particular, ( G, τ, ⊕ ) is of first category.Proof. It follows from Lemma 3.4 that the family ξ = { A ⊂ κ : Int( H A ) = ∅} is a freeultrafilter on κ . Since κ is a non-measurable cardinal, there is a family { B n : n ∈ N } such that B n ∈ ξ for every n and T { B n : n ∈ N } = ∅ . Then T { H B n : n ∈ N } = ∅ , thus S { H κ \ B n : n ∈ N } = G . Set A n = κ \ B n . Since A n ξ for every n ∈ N , it follows thatInt( H A n ) = ∅ . Above all, we know that { H A n : n ∈ N } is a family of nowhere densesets whose union covers G . The proof is completed. (cid:3) Corollary 3.6. If ( G, τ, ⊕ ) is a non-discrete irresolvable left topological gyrogroup (orright topological gyrogroup) of non-measurable cardinality, then ( G, τ, ⊕ ) is of first-category.Proof. Let ∆(
G, τ, ⊕ ) = κ . If κ = ω , it is obvious.Suppose that κ > ω . There exists an open neighborhood U of 0 such that | U | = κ .Then the gyrogroup G = h U i is open in G and the dispersion character of G coincideswith its power. However, it follows from Theorem 3.5 that G is of first category.Therefore, G is of first category. (cid:3) Corollary 3.7.
Every non-discrete irresolvable topological gyrogroup of non-measurablecardinality is of first category. A (strongly) σ -discrete space is one which is a countable union of (closed) discretesubspaces. Corollary 3.8.
Every submaximal topological gyrogroup of non-measurable cardinalityis strongly σ -discrete.Proof. It follows directly from the facts that every nowhere dense subset is closed anddiscrete in a submaximal space and every submaximal space is irresolvable. (cid:3) Submaximal properties of strongly topological gyrogroups
In this section, we prove that the cellularity of every submaximal strongly topologicalgyrogroup G is equal to the cardinality of G . Further, we prove that every submaximalstrongly topological gyrogroup of non-measurable cardinality is hereditarily paracom-pact.A topological gyrogroup ( G, τ, ⊕ ) is left κ -bounded for some cardinal κ if for everyopen neighborhood U of the element 0 there exists a subset A ⊂ G with | A | ≤ κ suchthat A ⊕ U = G . First, we need some lemmas in order to obtain one of main results inthis section. MENG BAO AND FUCAI LIN*
Lemma 4.1. [16]
Suppose that ( G, τ, ⊕ ) is a strongly topological gyrogroup with a sym-metric neighborhood base U at . Suppose further that U, V, W are all open neighbor-hoods of such that V ⊕ V ⊂ W , W ⊕ W ⊂ U and V, W ∈ U . If a subset A of G is U -disjoint, then the family of open sets { a ⊕ V : a ∈ A } is discrete in G . Lemma 4.2.
Let ( G, τ, ⊕ ) be a strongly topological gyrogroup with a symmetric openneighborhood base U at . If c ( G ) ≤ κ , then G is left κ -bounded.Proof. For an arbitrary open neighborhood U of the identity element 0 in G , there exist V, W ∈ U such that V ⊕ V ⊂ W and W ⊕ W ⊂ U . Let F = { A ⊂ G : ( b ⊕ V ) ∩ ( a ⊕ V ) = ∅ , for any distinct a, b ∈ A } . Define ≤ in G such that A ≤ A if and only if A ⊂ A , for any A , A ∈ F . Then,( F , ≤ ) is a poset and the union of any chain of V -disjoint sets is again a V -disjoint set.Therefore, it follows from Zorn’s Lemma that there exists a maximal element A in F so that { a ⊕ V : a ∈ A } is a disjoint family of non-empty spen sets in G . By Lemma4.1, the family of open sets { a ⊕ V : a ∈ A } is discrete in G . Since c ( G ) ≤ κ , it followsthat | A | ≤ κ .Since A is maximal, for every x ∈ G , there exists a ∈ A such that ( x ⊕ V ) ∩ ( a ⊕ V ) = ∅ .Then, there exist v , v ∈ V such that x ⊕ v = a ⊕ v . By the right cancellation law(2) in Lemma 2.3, we have that x = ( x ⊕ v ) ⊕ gyr [ x, v ]( ⊖ v )= ( a ⊕ v ) ⊕ gyr [ x, v ]( ⊖ v ) ∈ ( a ⊕ v ) ⊕ gyr [ x, v ]( V )= ( a ⊕ v ) ⊕ V = a ⊕ ( v ⊕ gyr [ v , a ]( V ))= a ⊕ ( v ⊕ V ) ⊂ a ⊕ ( V ⊕ V ) ⊂ a ⊕ U. Therefore, A ⊕ U = G . (cid:3) Corollary 4.3.
Every separable strongly topological gyrogroup G is left ω -narrow. From [15, Corollary 5.10], it follows that every pseudocompact rectifiable space is aSouslin space. Therefore, we have the following corollary.
Corollary 4.4.
Every pseudocompact strongly topological gyrogroup is left ω -narrow. Proposition 4.5.
If (
G, τ, ⊕ ) is a left κ -bounded strongly topological gyrogroup witha symmetric open neighborhood base U at 0 and H is a subgyrogroup of G , then H isalso left κ -bounded. Proof.
Let W be an arbitrary open neighborhood of 0 in H . Then we can fix V ∈ U such that ( V ⊕ V ) ∩ H ⊂ W . Since G is left κ -bounded, there is a set B with | B | ≤ κ in G such that B ⊕ V = G . Let C = { c ∈ B : ( c ⊕ V ) ∩ H = ∅} . It is obvious that | C | ≤ | B | ≤ κ and H ⊂ C ⊕ V . We can find a c ∈ ( c ⊕ V ) ∩ H for every c ∈ C . Let A = { a c : c ∈ C } and | A | ≤ κ in H . We show that A ⊕ W = H .Since H is a subgyrogroup and ( V ⊕ V ) ∩ H ⊂ W ⊂ H , we have that ( A ⊕ ( V ⊕ V )) ∩ H ⊂ A ⊕ W . Hence, it suffices to prove H ⊂ A ⊕ ( V ⊕ V ). For every c ∈ C , there exists UBMAXIMAL PROPERTIES IN (STRONGLY) TOPOLOGICAL GYROGROUPS 9 v ∈ V such that a c = c ⊕ v . Therefore c = ( c ⊕ v ) ⊕ gyr [ c, v ]( ⊖ v )= a c ⊕ gyr [ c, v ]( ⊖ v ) ∈ a c ⊕ gyr [ c, v ]( V )= a c ⊕ V. Thus, C ⊂ A ⊕ V . Moreover, since H ⊂ C ⊕ V , we have that H ⊂ ( A ⊕ V ) ⊕ V ⊂ A ⊕ ( V ⊕ gyr [ V, A ]( V ))= A ⊕ ( V ⊕ V ) . Therefore, H = A ⊕ W . (cid:3) Lemma 4.6.
Let ( G, τ, ⊕ ) be a strongly topological gyrogroup with a symmetric openneighborhood base U at and H a closed and discrete subgyrogroup of G . Take anyopen neighborhood V of the identity element in G such that V ∩ H = { } .(1) If U ∈ U such that U ⊕ U ⊂ V , then H A ⊕ U for any A ⊂ G with | A | < | H | .(2) If W, U ∈ U such that W ⊕ W ⊂ V and U ⊕ U ⊂ W , then the family { g ⊕ U : g ∈ H } is discrete in G .Proof. Indeed, if there is A ⊂ G with H ⊂ A ⊕ U such that | A | < | H | , then for some a ∈ A the set a ⊕ U must contain at least two elements of H . Take any h, g ∈ H ∩ ( a ⊕ U )such that h = g . Then there are v, u ∈ U such that h = a ⊕ u and g = a ⊕ v . Since a = ( a ⊕ v ) ⊕ gyr [ a, v ]( ⊖ v )= g ⊕ gyr [ a, v ]( ⊖ v ) ∈ g ⊕ gyr [ a, v ]( U )= g ⊕ U, it follows that h = a ⊕ u ∈ ( g ⊕ U ) ⊕ u ⊂ ( g ⊕ U ) ⊕ U = g ⊕ ( U ⊕ gyr [ U, g ]( U ))= g ⊕ ( U ⊕ U ) ⊂ g ⊕ V. Therefore, 0 = ⊖ g ⊕ h ∈ V ∩ H , which is a contradiction.In order to prove (2), we take an arbitrary g ∈ G . We show that g ⊕ U intersects atmost one element of { x ⊕ U : x ∈ H } . Suppose not, then we can find a, b, c, d ∈ U and distinct p, q ∈ H such that p ⊕ a = g ⊕ b and q ⊕ c = g ⊕ d . Since p = ( p ⊕ a ) ⊕ gyr [ p, a ]( ⊖ a )= ( g ⊕ b ) ⊕ gyr [ p, a ]( ⊖ a ) ∈ ( g ⊕ b ) ⊕ gyr [ p, a ]( U )= ( g ⊕ b ) ⊕ U ⊂ ( g ⊕ U ) ⊕ U = g ⊕ ( U ⊕ gyr [ U, g ]( U ))= g ⊕ ( U ⊕ U ) ⊂ g ⊕ W, and by the same method, we know that g ∈ q ⊕ W . Therefore, p ∈ g ⊕ W ⊂ ( q ⊕ W ) ⊕ W = q ⊕ ( W ⊕ gyr [ W, q ]( W )) = q ⊕ ( W ⊕ W ) ⊂ q ⊕ V, which implies that ( ⊖ q ) ⊕ p ∈ V . Since p and q are different, we have ( ⊖ q ) ⊕ p = 0 and( ⊖ q ) ⊕ p ∈ V ∩ H , which is a contradiction. (cid:3) Lemma 4.7. If G is a strongly topological gyrogroup, then there exists an open L -subgyrogroup H of G such that | H | = △ ( G ) .Proof. Let G be a strongly topological gyrogroup with a symmetric open neighborhoodbase U at 0. Since U is a base at 0 and G is homogeneous, it follows from Definition 2.11that there exists an open neighborhood V of 0 such that | V | = △ ( G ), then we can find U ∈ U such that U ⊂ V . Clearly, | U | ≤ | V | because U ⊂ V , hence | U | = △ ( G ). Put H = U . We define a sequence { H n } n ∈ N of subsets of G such that H n +1 = ⊖ ( H n ⊕ H n ) ∪ ( H n ⊕ H n ) for each n ∈ N . Set H = S n ∈ N H n , hence H is open in G and | H | = △ ( G ).We claim that H is an L -subgyrogroup in G . It is clear that H is closed for thegyrogroup operation and the inverse, so H is a subgyrogroup of G . Moreover, for any x, y ∈ G , gyr[ x, y ] is a groupoid homomorphism from G onto itself and gyr[ x, y ]( U ) = U .Next we claim that gyr[ x, y ]( H n ) = H n for any x, y ∈ G and n ∈ N . Clearly, we havegyr[ x, y ]( H ) = H for all x, y ∈ G . By induction, we may assume that for some n ∈ N we have gyr[ x, y ]( H n ) = H n for any x, y ∈ G . Now we prove gyr[ x, y ]( H n +1 ) = H n +1 for any x, y ∈ G .Indeed, for all x, y ∈ G , we havegyr[ x, y ]( H n +1 ) = gyr[ x, y ]( ⊖ ( H n ⊕ H n ) ∪ ( H n ⊕ H n ))= ⊖ (gyr[ x, y ]( H n ) ⊕ gyr[ x, y ]( H n )) ∪ (gyr[ x, y ]( H n ) ⊕ gyr[ x, y ]( H n ))= ⊖ ( H n ⊕ H n ) ∪ ( H n ⊕ H n )= H n +1 . Then, for any z ∈ H , there exists n ∈ N such that z ∈ H n . It follows that gyr[ x, y ]( z ) ∈ gyr[ x, y ]( H n ) = H n ⊂ H . Hence gyr[ x, y ]( H ) = H . Therefore, H is an L -subgyrogroupof G . (cid:3) Let the gyrogroup K be endowed with discrete topology [23, p. 41] and let D betopological gyrogroup in Example 2.13. Put G = K × D , where G is endowed with theproduct topology and the operation with coordinate. Fix an arbitrary L -subgyrogroup H in K , for example, H = { , , , } or { , , , , · · · , } . Then H is an open L -subgyrogroup of G such that | H | = △ ( G ). Theorem 4.8.
Let κ be an infinite cardinal number. If G is a left κ -bounded submaximalstrongly topological gyrogroup, then | G | ≤ κ . UBMAXIMAL PROPERTIES IN (STRONGLY) TOPOLOGICAL GYROGROUPS 11
Proof.
Let △ ( G ) = λ , and from Lemma 4.7 take an open L -subgyrogroup N of G suchthat | N | = λ . Then G is a discrete union of left translations of N and it is impossibleto cover G by less than | G/N | left translations of N . Thus, | G/N | ≤ κ . Then it sufficesto prove λ = | N | ≤ κ . We divide the proof into the following two cases. Case 1 λ is a limit cardinal.We assume that κ < λ and take a subset P ⊂ N such that κ < | P | = γ < λ . Then |h P i| = γ and H = h P i is closed and discrete in G since G is submaximal. From (1) ofLemma 4.6, it follows that G is not left κ -bounded, which is a contradiction. Case 2 λ is a successor cardinal.Then λ = cf ( λ ). Since the L -subgyrogroup N is submaximal, it follows that △ ( N ) = | N | , then we can take a canonical decomposition { H α : α < λ } for N . By Lemma 3.4,there exists a cofinal set A ⊂ λ such that H A = S { H α : α ∈ A } is closed and discretein G and 0 H A .For every α ∈ A , choose a point x α ∈ H α . It is obvious that Y = { x α : α ∈ A } isclosed and discrete in G . By the confinality of A in λ , we have | Y | = λ . Hence we canfind an open neighborhood U of 0 such that U ∩ H A = ∅ . There exists V ∈ U suchthat V ⊕ V ⊂ U . We claim that P ⊕ V = G for any P ⊂ G with | P | < λ. Suppose not, then there exists P ⊂ G with | P | < λ such that P ⊕ V = G . Since | Y | = λ and | P | < λ , there exists a p ∈ P such that p ⊕ V contains at least twodistinct points x α , x β of Y , where α < β . It follows from ⊖ x α ∈ H α and x β ∈ H β that( ⊖ x α ) ⊕ x β ∈ H β ⊂ H A by (6) of Lemma 3.3. Moreover, we can find u, v ∈ V such that p ⊕ u = x α and p ⊕ v = x β . Since p = ( p ⊕ u ) ⊕ gyr [ p, u ]( ⊖ u ) = x α ⊕ gyr [ p, u ]( ⊖ u ) ∈ x α ⊕ gyr [ p, u ]( V ) = x α ⊕ V, then x β ∈ ( x α ⊕ V ) ⊕ V = x α ⊕ ( V ⊕ gyr [ V, x α ]( V )) = x α ⊕ ( V ⊕ V ) . Thus, ⊖ x α ⊕ x β ∈ ( V ⊕ V ) ∩ H A ⊂ U ∩ H A , which is a contradiction. Thus, P ⊕ V = G .Therefore, κ ≥ λ . (cid:3) Now we can easily obtain the first main result in this section.
Theorem 4.9. c ( G ) = | G | for every submaximal strongly topological gyrogroup G .In particular, a submaximal strongly topological gyrogroup with the Suslin property iscountable.Proof. By Lemma 4.2, if c ( G ) ≤ κ , we have that G is left κ -bounded. Then it followsfrom Theorem 4.8 that | G | ≤ κ . (cid:3) Finally, we prove the second main result in this section.
Lemma 4.10. [3, Lemma 3.13]
Let X be a regular space. Suppose that X = S { H n : n ∈ N } , where the subsets H n have the following properties:(1) H i is closed and discrete in X for all i ∈ N ;(2) H i ∩ H j = ∅ if i = j ;(3) for every x ∈ X there is an open neighborhood V x of x such that for any i ∈ N the family { V x : x ∈ H i } is discrete in X .Then X is weakly collectionwise Hausdorff. Theorem 4.11.
Let ( G, τ, ⊕ ) be a submaximal strongly topological gyrogroup with asymmetric open neighborhood base U at . If G has non-measurable cardinality, then G is hereditarily paracompact. Proof.
From Lemma 4.7, G has an open L -subgyrogroup whose cardinality and disper-sion character are equal. Hence, if we prove hereditary paracompactness of this open L -subgyrogroup of G , then G will be hereditarily paracompact. Without loss of gener-ality, we may assume that | G | = κ is a uncountable cardinal such that △ ( G ) = κ . From[3, Theorem 2.2], each submaximal weakly collectionwise Hausdorff space is hereditar-ily paracompact, hence it suffices to prove that G is weakly collectionwise Hausdorff.Take a canonical decomposition { H α : α < κ } of G . We verify that a family of subsets { P n : n ∈ N } satisfies the conditions of Lemma 4.10.Indeed, it follows from Theorem 3.5 that there exists a family { A n : n ∈ N } of subsetsof κ such that S { A n : n ∈ N } = κ and P n = H A n = S { H α : α ∈ A n } is closed anddiscrete in G . Now we only need to check the condition (3) in Lemma 4.10.For each n ∈ N there exists an open neighborhood U n of 0 with U n ∩ H A n ⊂ { } .Choose V n , W n ∈ U such that V n ⊕ V n ⊂ W n and W n ⊕ W n ⊂ U n . For every α ∈ A n ,let W nα = H α ⊕ V n . For arbitrary g ∈ G , we show that O = g ⊕ V n can intersect atmost one element of γ n = { W nα : α ∈ A n } . We assume that there are α, β ∈ A n with α < β such that W nα ∩ O = ∅ 6 = W nβ ∩ O . Therefore, there are p ∈ H α , q ∈ H β and u, v, u , v ∈ V n such that g ⊕ v = p ⊕ v and g ⊕ u = q ⊕ u . Since g = ( g ⊕ u ) ⊕ gyr [ g, u ]( ⊖ u )= ( q ⊕ u ) ⊕ gyr [ g, u ]( ⊖ u ) ∈ ( q ⊕ u ) ⊕ gyr [ g, u ]( V n ) ⊂ ( q ⊕ V n ) ⊕ V n = q ⊕ ( V n ⊕ gyr [ V n , q ]( V n )= q ⊕ ( V n ⊕ V n ) ⊂ q ⊕ W n . Then, by the same method, we have p ∈ g ⊕ W n ⊂ ( q ⊕ W n ) ⊕ W n = q ⊕ ( W n ⊕ gyr [ W n , q ]( W n ))= q ⊕ ( W n ⊕ W n ) ⊂ q ⊕ U n . Therefore, 0 = ( ⊖ q ) ⊕ p ∈ U n . Moreover, ⊖ q ∈ H β implies ( ⊖ q ) ⊕ p ∈ H β by (6) ofLemma 3.3. It is contradict with H A n ∩ U n ⊂ { } . Thus, γ n = { W nα : α ∈ A n } isdiscrete in G .For each α ∈ A n , since | G α | < κ , we have that the subgyrogroup G α = S { H υ : υ ≤ α } is closed and discrete in G . Let U be an open neighborhood of 0 such that U ∩ G α = { } .We can find V, W ∈ U such that V ⊕ V ⊂ W and W ⊕ W ⊂ U . It follows from (2) ofLemma 4.6 that the family µ = { x ⊕ V : x ∈ G α } is discrete.For an arbitrary n ∈ N , if α ∈ A n and p ∈ H α , let V p = ( p ⊕ V ) ∩ W nα . It is clear that µ n = { V p : p ∈ P n } is discrete in G and p ∈ V p for every p ∈ P n .Therefore, the conditions (1)-(3) in Lemma 4.10 hold and it follows from Lemma 4.10that G is weakly collectionwise Hausdorff. (cid:3) By [10, Theorem 7.2], we know that if a normal space is a countable union of itsstrongly zero-dimensional spaces, then it is strongly zero-dimensional. Therefore, wehave the following results.
UBMAXIMAL PROPERTIES IN (STRONGLY) TOPOLOGICAL GYROGROUPS 13
Corollary 4.12.
If a submaximal strongly topological gyrogroup G has non-measurablecardinality, then dim ( G ) = 0 . In particular, G cannot be connected. Corollary 4.13.
If there does not exist any measurable cardinal, then every submaximalstrongly topological gyrogroup is hereditarily paracompact and zero-dimensional in thesense of the dimension dim. In particular, no submaximal infinite strongly topologicalgyrogroup is connected.
Since every strongly topological gyrogroup is a topological gyrogroup, it is natural topose the following question.
Question 4.14.
Is each submaximal topological gyrogroup G of non-measurable cardi-nality hereditarily paracompact? Acknowledgements . We wish to thank anonymous referees for the detailed list ofcorrections, suggestions to the paper, and all her/his efforts in order to improve thepaper.
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Email address : [email protected] (Fucai Lin): School of mathematics and statistics, Minnan Normal University, Zhangzhou363000, P. R. China Email address ::