Strongly Topological Gyrogroups and Quotient With Respect to L-subgyrogroups
aa r X i v : . [ m a t h . GN ] F e b STRONGLY TOPOLOGICAL GYROGROUPS AND QUOTIENTWITH RESPECT TO L -SUBGYROGROUPS MENG BAO, XUEWEI LING, AND XIAOQUAN XU*
Abstract.
In this paper, some generalized metric properties in strongly topologicalgyrogroups are studied. In particular, it is proved that when G is a strongly topologicalgyrogroup with a symmetric neighborhood base U at 0 and H is a second-countableadmissible subgyrogroup generated from U , if the quotient space G/H is an ℵ -space(resp., cosmic space), then G is also an ℵ -space (resp., cosmic space); If the quotientspace G/H has a star-countable cs -network (resp., wcs ∗ -network, k -network), then G also has a star-countable cs -network (resp., wcs ∗ -network, k -network). Moreover, itis shown that when G is a strongly topological gyrogroup with a symmetric neigh-borhood base U at 0 and H is a locally compact metrizable admissible subgyrogroupgenerated from U , if the quotient space G/H is sequential, then G is also sequential;Furthermore, if the quotient space G/H is strictly (strongly) Fr´echet-Urysohn, then G is also strictly (strongly) Fr´echet-Urysohn; Finally, if the quotient space G/H is astratifiable space (semi-stratifiable space, σ -space, k -semistratifiable space), then G isa local stratifiable space (semi-stratifiable space, σ -space, k -semistratifiable space). Introduction
The c -ball of relativistically admissible velocities with the Einstein velocity additionwas researched for many years. The Einstein velocity 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 . In particular, by the research of this, Ungar in [40, 41] posed the concept of a gy-rogroup. It is obvious that a gyrogroup has a weaker algebraic structure than a group.Then, in 2017, a gyrogroup was endowed with a topology by Atiponrat [2] such thatthe multiplication is jointly continuous and the inverse is also continuous. At the sametime, she claimed that M¨obius gyrogroups, Einstein gyrogroups, and Proper veloc-ity gyrogroups, that were studied in [14, 15, 16, 41], are all topological gyrogroups.Moreover, Cai, Lin and He in [10] proved that every topological gyrogroup is a rec-tifiable space and deduced that first-countability and metrizability are equivalent intopological gyrogroups. Indeed, this kind of space has been studied for many years, see[3, 4, 22, 24, 25, 26, 36, 37, 38, 39, 42, 43, 44]. After then, in 2019, Bao and Lin [5] defined
Mathematics Subject Classification.
Primary 54A20; secondary 11B05; 26A03; 40A05; 40A30;40A99.
Key words and phrases.
Strongly topological gyrogroups; Fr´echet-Urysohn; networks; stratifiablespaces; k -semistratifiable spaces.The authors are supported by the National Natural Science Foundation of China (Nos. 11661057,12071199) and the Natural Science Foundation of Jiangxi Province, China (No. 20192ACBL20045)*corresponding author. the concept of strongly topological gyrogroups and claimed that M¨obius gyrogroups,Einstein gyrogroups, and Proper velocity gyrogroups endowed with standard topologyare all strongly topological gyrogroups but not topological groups. Furthermore, theyproved that every strongly topological gyrogroup with a countable pseudocharacter issubmetrizable and every locally paracompact strongly topological gyrogroup is para-compact [6, 7]. They also claimed that every feathered strongly topological gyrogroupis paracompact, and hence a D -space [5]. In the same paper, they gave an exampleto show that there exists a strongly topological gyrogroup which has an infinite L -subgyrogroup. Therefore, it is meaningful to research the quotient spaces of a stronglytopological gyrogroup with respect to L -subgyrogroups as left cosets. In particular,we investigate what properties of topological groups still valid on strongly topologicalgyrogroups.In this paper, we mainly study some generalized metric properties in strongly topo-logical gyrogroups. In Section 3, it is proved that when G is a strongly topologicalgyrogroup with a symmetric neighborhood base U at 0 and H is a second-countableadmissible subgyrogroup generated from U , if the quotient space G/H is an ℵ -space(resp., cosmic space), then G is also an ℵ -space (resp., cosmic space); If the quotientspace G/H has a star-countable cs -network (resp., wcs ∗ -network, k -network), then G also has a star-countable cs -network (resp., wcs ∗ -network, k -network). In Section 4, weinvestigate the quotient space G/H with some generalized metric properties when G isa strongly topological gyrogroup with a symmetric neighborhood base U at 0 and H is a locally compact metrizable admissible subgyrogroup generated from U . We showthat when G is a strongly topological gyrogroup with a symmetric neighborhood base U at 0 and H is a locally compact metrizable admissible subgyrogroup generated from U , if the quotient space G/H is sequential, then G is also sequential; Furthermore, ifthe quotient space G/H is strictly (strongly) Fr´echet-Urysohn, then G is also strictly(strongly) Fr´echet-Urysohn; Finally, if the quotient space G/H is a stratifiable space(semi-stratifiable space, σ -space, k -semistratifiable space), then G is a local stratifiablespace (semi-stratifiable space, σ -space, k -semistratifiable space).2. Preliminaries
Throughout this paper, all topological spaces are assumed to be Hausdorff, unlessotherwise is explicitly stated. Let N be the set of all positive integers and ω the firstinfinite ordinal. The readers may consult [1, 12, 30, 41] for notation and terminologynot explicitly given here. Next we recall some definitions and facts. Definition 2.1. [41] 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 ].Notice that a group is a gyrogroup ( G, ⊕ ) such that gyr[ x, y ] is the identity functionfor all x, y ∈ G . TRONGLY TOPOLOGICAL GYROGROUPS AND QUOTIENT WITH RESPECT TO L -SUBGYROGROUPS3 Lemma 2.2. [41]
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 )) . (5) ( x ⊕ y ) ⊕ z = x ⊕ ( y ⊕ gyr [ y, x ]( z )) . The definition of a subgyrogroup is given as follows.
Definition 2.3. [37] 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 .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 [37], 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 , whichexplains that by the item (4) in Lemma 2.2 it follows the subgyrogroup criterion. Definition 2.4. [2] 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.Obviously, every topological group is a topological gyrogroup. However, any topo-logical gyrogroup whose gyrations are not identically equal to the identity is not atopological group. Definition 2.5. [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 at 0.Clearly, we may assume that U is symmetric for each U ∈ U in Definition 2.5.Moreover, in the classical M¨obius, Einstein, or Proper Velocity gyrogroups we know thatgyrations are indeed special rotations, however for an arbitrary gyrogroup, gyrationsbelong to the automorphism group of G and need not be necessarily rotations.In [5], the authors proved that there is a strongly topological gyrogroup which is nota topological group, see Example 2.6. Example 2.6. [5]
Let D be the complex open unit disk { z ∈ C : | z | < } . We consider D with the standard topology. In [2, 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 [2, Example 2] that gyr [ a, b ]( c ) = 1 + a ¯ b ab c for any a, b, c ∈ D . MENG BAO, XUEWEI LING, AND XIAOQUAN XU*
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 [37, Proposition 2.6] . Hence, ( D , ⊕ M ) is a strongly topological gyrogroup. However, ( D , ⊕ M ) is not a group [2, Example 2] . Remark 1.
Even though M¨obius gyrogroups, Einstein gyrogroups, and Proper ve-locity gyrogroups are all strongly topological gyrogroups, all of them do not possessany non-trivial L -subgyrogroups. However, there is a class of strongly topological gy-rogroups which has a non-trivial L -subgyrogroup, see the following example. Example 2.7. [5]
There exists a strongly topological gyrogroup which has an infinite L -subgyrogroup. Indeed, let X be an arbitrary feathered non-metrizable topological group, and let Y be an any strongly topological gyrogroup with a non-trivial L -subgyrogroup (such asthe gyrogroup K [43, p. 41]). Put G = X × Y with the product topology and theoperation with coordinate. Then G is an infinite strongly topological gyrogroup since X is infinite. Let H be a non-trivial L -subgyrogroup of Y , and take an arbitrary infinitesubgroup N of X . Then N × H is an infinite L -subgyrogroup of G . Definition 2.8. [12, 19, 21, 27] Let P be a family of subsets of a topological space X .1. P is called a network for X if whenever x ∈ U with U open in X , then there exists P ∈ P such that x ∈ P ⊂ U .2. P is called a k-network for X if whenever K ⊂ U with K compact and U open in X , there exists a finite family P ′ ⊂ P such that K ⊂ S P ′ ⊂ U .3. P is called a cs-network for X if, given a sequence { x n } n converging to a point x in X and a neighborhood U of x in X , then { x } ∪ { x n : n ≥ n } ⊂ P ⊂ U for some n ∈ N and some P ∈ P .4. P is called a wcs ∗ -network for X if, given a sequence { x n } n converging to a point x in X and a neighborhood U of x in X , there exists a subsequence { x n i } i of the sequence { x n } n such that { x n i : i ∈ N } ⊂ P ⊂ U for some P ∈ P .It is claimed in [30] that every base is a k -network and a cs -network for a topologicalspace, and every k -network or every cs -network is a wcs ∗ -network for a topological space,but the converse does not hold. Moreover, a space X has a countable cs -network if andonly if X has a countable k -network if and only if X has a countable wcs ∗ -network, see[29]. Definition 2.9. [33] Let X be a topological space.1. X is called cosmic if X is a regular space with a countable network.2. X is called an ℵ -space if it is a regular space with a countable k -network. Remark 2.
It was shown in [18] that every separable metric space is an ℵ -space.Moreover, every ℵ -space is a cosmic space and every cosmic space is a paracompact,separable space. Then, in [21], it was proved that a topological space is an ℵ -space ifand only if it is a regular space with a countable cs -network.Now we recall the following concept of the coset space of a topological gyrogroup.Let ( G, τ, ⊕ ) be a topological gyrogroup and H an L -subgyrogroup of G . It followsfrom [37, Theorem 20] that G/H = { a ⊕ H : a ∈ G } is a partition of G . We denoteby π the mapping a a ⊕ H from G onto G/H . Clearly, for each a ∈ G , we have TRONGLY TOPOLOGICAL GYROGROUPS AND QUOTIENT WITH RESPECT TO L -SUBGYROGROUPS5 π − { π ( a ) } = a ⊕ H . Denote by τ ( G ) the topology of G . In the set G/H , we define afamily τ ( G/H ) of subsets as follows: τ ( G/H ) = { O ⊂ G/H : π − ( O ) ∈ τ ( G ) } . The following concept of an admissible subgyrogroup of a strongly topological gy-rogroup was first introduced in [6], which plays an important role in this paper.A subgyrogroup H of a topological gyrogroup G is called admissible if there existsa sequence { U n : n ∈ ω } of open symmetric neighborhoods of the identity 0 in G suchthat U n +1 ⊕ ( U n +1 ⊕ U n +1 ) ⊂ U n for each n ∈ ω and H = T n ∈ ω U n . If G is a stronglytopological gyrogroup with a symmetric neighborhood base U at 0 and each U n ∈ U ,we say that the admissible topological subgyrogroup is generated from U [7].It was shown in [7] that if G is a strongly topological gyrogroup with a symmetricneighborhood base U at 0, then each admissible topological subgyrogroup H generatedfrom U is a closed L -subgyrogroup of G .3. Quotient with respect to second-countable admissible subgyrogroups
In this section, we study the quotient space
G/H with some generalized metric prop-erties when G is a strongly topological gyrogroup with a symmetric neighborhood base U at 0 and H is a second-countable admissible subgyrogroup of G generated from U .Suppose that G is a strongly topological gyrogroup with a symmetric neighborhoodbase U at 0 and H is a second-countable admissible subgyrogroup generated from U .We prove that if the quotient space G/H is an ℵ -space (resp., cosmic space), then G is also an ℵ -space (resp., cosmic space). Moreover, we show that if the quotient space G/H has a star-countable cs -network (resp., wcs ∗ -network, k -network), then G also hasa star-countable cs -network (resp., wcs ∗ -network, k -network). Lemma 3.1. [5]
Let ( G, τ, ⊕ ) be a topological gyrogroup and H an L -subgyrogroup of G . Then the natural homomorphism π from a topological gyrogroup G to its quotienttopology on G/H is an open and continuous mapping.
Lemma 3.2. [1]
Suppose that f : X → Y is an open continuous mapping of a space X onto a space Y , x ∈ X , B ⊂ Y , and f ( x ) ∈ B . Then x ∈ f − ( B ) . In particular, f − ( B ) = f − ( B ) . Proposition 3.3.
Suppose that G is a topological gyrogroup and H is a closed andseparable L -subgyrogroup of G . If Y is a separable subset of G/H , π − ( Y ) is alsoseparable in G . Proof.
Let π be the natural homomorphism from G onto the quotient space G/H . Since Y is a separable subset of G/H , there is a countable subset B of G/H such that Y ⊂ B .For each y ∈ B , we can find x ∈ G such that π ( x ) = y . Since H is separable and π − ( π ( x )) = x ⊕ H is homeomorphic to H , there is a countable subset M y of π − ( π ( x ))such that M y = x ⊕ H . Put M = S { M y : y ∈ B } . It is clear that M is countable and M = π − ( B ). It follows from Lemma 3.1 that π is an open and continuous mapping.Then, π − ( Y ) ⊂ π − ( B ) = π − ( B ) = M = M by Lemma 3.2. Therefore, π − ( Y ) isseparable in G . (cid:3) Lemma 3.4. [7]
Every locally paracompact strongly topological gyrogroup is paracom-pact.
Lemma 3.5. [9]
Every star-countable family P of subsets of a topological space X can be expressed as P = S {P α : α ∈ Λ } , where each subfamily P α is countable and ( S P α ) ∩ ( S P β ) = ∅ whenever α = β . MENG BAO, XUEWEI LING, AND XIAOQUAN XU*
Theorem 3.6.
Let G be a strongly topological gyrogroup with a symmetric neighborhoodbase U at and let H be a second-countable admissible subgyrogroup generated from U . If the quotient space G/H is a local ℵ -space (resp., locally cosmic space), then G is a topological sum of ℵ -subspace (resp., cosmic subspaces).Proof. We just prove the case of ℵ -space, and the case of cosmic space is similar.Let U be a symmetric neighborhood base at 0 such that gyr [ x, y ]( U ) = U for any x, y ∈ G and U ∈ U . Since H is an admissible subgyrogroup generated from U , thereexits a sequence { U n : n ∈ ω } of open symmetric neighborhoods of the identity 0 in G such that U n ∈ U , U n +1 ⊕ ( U n +1 ⊕ U n +1 ) ⊂ U n for each n ∈ ω and H = T n ∈ ω U n .By the hypothesis, the quotient space G/H is a local ℵ -space. Then we can find anopen neighborhood Y of H in G/H such that Y has a countable cs -network. Put X = π − ( Y ). Since the natural homomorphism π from G onto G/H is an open andcontinuous mapping, X is an open neighborhood of the identity element 0 in G . Itfollows from Proposition 3.3 that X is separable. Therefore, there is countable subset B = { b m : m ∈ N } of X such that B = X .By the first-countability of H , there exists a countable family { V n : n ∈ N } ⊂ U ofopen symmetric neighborhoods of 0 in G such that V n +1 ⊕ ( V n +1 ⊕ V n +1 ) ⊂ V n ⊂ X foreach n ∈ N and the family { V n ∩ H : n ∈ N } is a local base at 0 for H . Since Y is an ℵ -space, there is a countable cs -network {P k : k ∈ N } for Y . Claim 1. X is an ℵ -space.Put F = { π − ( P k ) ∩ ( b m ⊕ V n ) : k, m, n ∈ N } . Then F is a countable family ofsubsets of X . Suppose that { x i } i is a sequence converging to a point x in X and U bea neighborhood of x in X . Then U is also a neighborhood of x in G . Let V be an openneighborhood of 0 in G such that x ⊕ ( V ⊕ V ) ⊂ U . Since { V n ∩ H : n ∈ N } is a localbase at 0 for H , there is n ∈ N such that V n ∩ H ⊂ V ∩ H . Moreover, ( x ⊕ V n +1 ) ∩ X is anon-empty open subset of X and B = X , whence B ∩ ( x ⊕ V n +1 ) = ∅ . Therefore, thereexists b m ∈ B such that b m ∈ x ⊕ V n +1 . Furthermore, ( x ⊕ V n +1 ) ∩ ( x ⊕ V ) is an openneighborhood of x and π : G → G/H is an open mapping, so π (( x ⊕ V n +1 ) ∩ ( x ⊕ V ))is an open neighborhood of π ( x ) in the space Y and the sequence { π ( x i ) } i converges to π ( x ) in Y . It is obtained that { π ( x ) } ∪ { π ( x i ) : i ≥ i } ⊂ P k ⊂ π (( x ⊕ V n +1 ) ∩ ( x ⊕ V )) f or some i , k ∈ N . Subclaim 1. ( x ⊕ V n +1 ) ∩ ( x ⊕ V ) = x ⊕ ( V n +1 ∩ V ).For every t ∈ ( x ⊕ V n +1 ) ∩ ( x ⊕ V ), there are u ∈ V n +1 , v ∈ V such that t = x ⊕ u = x ⊕ v . By Lemma 2.2, u = ⊖ x ⊕ ( x ⊕ u ) = ⊖ x ⊕ ( x ⊕ v ) = v . Therefore,( x ⊕ V n +1 ) ∩ ( x ⊕ V ) ⊂ x ⊕ ( V n +1 ∩ V ).On the contrary, for any s ∈ x ⊕ ( V n +1 ∩ V ), there is u ∈ V n +1 ∩ V such that s = x ⊕ u .It is obvious that s ∈ ( x ⊕ V n +1 ) ∩ ( x ⊕ V ), that is, x ⊕ ( V n +1 ∩ V ) ⊂ ( x ⊕ V n +1 ) ∩ ( x ⊕ V ).Hence, ( x ⊕ V n +1 ) ∩ ( x ⊕ V ) = x ⊕ ( V n +1 ∩ V ). Subclaim 2. π − ( P k ) ∩ ( b m ⊕ V n +1 ) ⊂ U .For an arbitrary z ∈ π − ( P k ) ∩ ( b m ⊕ V n +1 ), π ( z ) ∈ P k ⊂ π (( x ⊕ V n +1 ) ∩ ( x ⊕ V )).Then, since z ∈ (( x ⊕ V n +1 ) ∩ ( x ⊕ V )) ⊕ H = ( x ⊕ ( V n +1 ∩ V )) ⊕ H , and H is anadmissible subgyrogroup generated from U , we have z ∈ ( x ⊕ ( V n +1 ∩ V )) ⊕ H = x ⊕ (( V n +1 ∩ V ) ⊕ gyr [( V n +1 ∩ V ) , x ]( H ))= x ⊕ (( V n +1 ∩ V ) ⊕ gyr [( V n +1 ∩ V ) , x ]( \ m ∈ N U m )) TRONGLY TOPOLOGICAL GYROGROUPS AND QUOTIENT WITH RESPECT TO L -SUBGYROGROUPS7 ⊂ x ⊕ (( V n +1 ∩ V ) ⊕ \ m ∈ N gyr [( V n +1 ∩ V ) , x ]( U m ))= x ⊕ (( V n +1 ∩ V ) ⊕ \ m ∈ N U m )= x ⊕ (( V n +1 ∩ V ) ⊕ H ) . Therefore, ⊖ x ⊕ z ∈ ( V n +1 ∩ V ) ⊕ H . Moreover, since z ∈ b m ⊕ V n +1 and b m ∈ x ⊕ V n +1 ,it follows that z ∈ ( x ⊕ V n +1 ) ⊕ V n +1 = x ⊕ ( V n +1 ⊕ gyr [ V n +1 , x ]( V n +1 ))= x ⊕ ( V n +1 ⊕ V n +1 ) . So, ( ⊖ x ) ⊕ z ∈ V n +1 ⊕ V n +1 . Hence, ( ⊖ x ) ⊕ z ∈ (( V n +1 ∩ V ) ⊕ H ) ∩ ( V n +1 ⊕ V n +1 ). Thereexist a ∈ ( V n +1 ∩ V ) , h ∈ H and u , v ∈ V n +1 such that ( ⊖ x ) ⊕ z = a ⊕ h = u ⊕ v ,whence h = ( ⊖ a ) ⊕ ( u ⊕ v ) ∈ V n +1 ⊕ ( V n +1 ⊕ V n +1 ) ⊂ V n . Therefore, ( ⊖ x ) ⊕ z ∈ ( V n +1 ∩ V ) ⊕ ( V n ∩ H ), and consequently, z ∈ x ⊕ (( V n +1 ∩ V ) ⊕ ( V n ∩ H )) ⊂ x ⊕ ( V ⊕ V ) ⊂ U .Since b m ∈ x ⊕ V n +1 , there is u ∈ V n +1 such that b m = x ⊕ u , whence x = ( x ⊕ u ) ⊕ gyr [ x, u ]( ⊖ u )= b m ⊕ gyr [ x, u ]( ⊖ u ) ∈ b m ⊕ gyr [ x, u ]( V n +1 )= b m ⊕ V n +1 . Therefore, there exists i ≥ i such that x i ∈ b m ⊕ V n +1 when i ≥ i , whence { x } ∪ { x i : i ≥ i } ⊂ π − ( P k ) ∩ ( b m ⊕ V n +1 ). Thus F is a countable cs -network for X , and wecomplete the proof of Claim 1.Since strongly topological gyrogroup G is homogeneous, G is a local ℵ -space byClaim 1. Therefore, G is a locally paracompact space. Furthermore, since every locallyparacompact strongly topological gyrogroup is paracompact by Lemma 3.4, G is para-compact. Let A is an open cover of G by ℵ -subspace. Because the property of beingan ℵ -space is hereditary, we can assume that A is locally finite in G by the paracom-pactness of G . Moreover, as every point-countable family of open subsets in a separablespace is countable, the family A is star-countable. Then A = S {B α : α ∈ Λ } by Lemma3.5, where each subfamily B α is countable and ( S B α ) ∩ ( S B β ) = ∅ whenever α = β .Set X α = S B α for each α ∈ Λ. Then G = L α ∈ Λ X α . Claim 2. X α is an ℵ -subspace for each α ∈ Λ.Put B α = { B α,n : n ∈ N } , where each B α,n is an open ℵ -subspace of G , and put P α = S n ∈ N P α,n , where P α,n is a countable cs -network for the ℵ -space B α,n for each n ∈ N . Then P α is a countable cs -network for X α . Thus, X α is an ℵ -space.In conclusion, the strongly topological gyrogroup G is a topological sum of ℵ -subspaces. (cid:3) Corollary 3.7.
Let G be a strongly topological gyrogroup with a symmetric neighborhoodbase U at and let H be a second-countable admissible subgyrogroup generated from U .If the quotient space G/H is an ℵ -space (resp., cosmic space), G is also an ℵ -space(resp., cosmic space). Theorem 3.8.
Let G be a strongly topological gyrogroup with a symmetric neighborhoodbase U at and let H be a second-countable admissible subgyrogroup generated from U . MENG BAO, XUEWEI LING, AND XIAOQUAN XU*
If the quotient space
G/H has a star-countable cs -network, G also has a star-countable cs -network.Proof. Let U be a symmetric neighborhood base at 0 such that gyr [ x, y ]( U ) = U forany x, y ∈ G and U ∈ U . Since H is an admissible subgyrogroup generated from U ,there exits a sequence { U n : n ∈ ω } of open symmetric neighborhoods of the identity 0in G such that U n ∈ U , U n +1 ⊕ ( U n +1 ⊕ U n +1 ) ⊂ U n for each n ∈ ω and H = T n ∈ ω U n .Since the L -subgyrogroup H of G is first-countable at the identity element 0 of G , thereexists a countable family { V n : n ∈ N } ⊂ U such that ( V n +1 ⊕ ( V n +1 ⊕ V n +1 )) ⊂ V n foreach n ∈ N and the family { V n ∩ H : n ∈ N } is a local base at 0 for H .Let P = { P α : α ∈ Λ } be a star-countable cs -network for the space G/H . For each α ∈ Λ, the family { P α ∩ P β : β ∈ Λ } is a countable wcs ∗ -network for P α . Therefore, P α is a cosmic space, and P α is separable. Then it follows from Proposition 3.3 that π − ( P α ) is separable. We can find a countable subset B α = { b α,m : m ∈ N } of π − ( P α )such that B α = π − ( P α ).Put F = { π − ( P α ) ∩ ( b α,m ⊕ V n ) : α ∈ Λ , and m, n ∈ N } . Then F is a star-countable family of G . Claim. F is a cs -network for G .Let { x i } i be a sequence converging to a point x in G and let U be a neighborhood of x in G . Choose an open neighborhood V of 0 in G such that ( x ⊕ ( V ⊕ V )) ⊂ U . Since { V n ∩ H : n ∈ N } is a local base at 0 for H , there exists n ∈ N such that V n ∩ H ⊂ V ∩ H .Since π : G → G/H is an open and continuous mapping, there are i ∈ N and α ∈ Λsuch that { π ( x ) } ∪ { π ( x i ) : i ≥ i } ⊂ P α ⊂ π (( x ⊕ V n +1 ) ∩ ( x ⊕ V )). Since x ∈ π − ( P α ),( x ⊕ V n +1 ) ∩ π − ( P α ) is non-empty and open in the subspace π − ( P α ). Moreover, since B α = π − ( P α ), there exists m ∈ N such that b α,m ∈ x ⊕ V n +1 . Subclaim. π − ( P α ) ∩ ( b α,m ⊕ V n +1 ) ⊂ U .For an arbitrary z ∈ π − ( P α ) ∩ ( b α,m ⊕ V n +1 ), π ( z ) ∈ P α ⊂ π (( x ⊕ V n +1 ) ∩ ( x ⊕ V )).By the proof of Theorem 3.6, z ∈ x ⊕ (( V n +1 ∩ V ) ⊕ H ). Since z ∈ b α,m ⊕ V n +1 and b α,m ∈ x ⊕ V n +1 , we have z ∈ ( x ⊕ V n +1 ) ⊕ V n +1 = x ⊕ ( V n +1 ⊕ gyr [ V n +1 , x ]( V n +1 ))= x ⊕ ( V n +1 ⊕ ( V n +1 )) . Then, ( ⊖ x ) ⊕ z ∈ V n +1 ⊕ V n +1 . Hence, ( ⊖ x ) ⊕ z ∈ (( V n +1 ∩ V ) ⊕ H ) ∩ ( V n +1 ⊕ V n +1 ).Therefore, there exist a ∈ ( V n +1 ∩ V ) , h ∈ H and u , u ∈ V n +1 such that ( ⊖ x ) ⊕ z = a ⊕ h = u ⊕ u , whence h = ( ⊖ a ) ⊕ ( u ⊕ u ) ∈ V n +1 ⊕ ( V n +1 ⊕ V n +1 ) ⊂ V n . It follows that( ⊖ x ) ⊕ z ∈ ( V n +1 ∩ V ) ⊕ ( V n ∩ H ). Thus z ∈ x ⊕ (( V n +1 ∩ V ) ⊕ ( V n ∩ H )) ⊂ x ⊕ ( V ⊕ V ) ⊂ U .Since b α,m ∈ x ⊕ V n +1 , there is u ∈ V n +1 such that b α,m = x ⊕ u . Thus, x = ( x ⊕ u ) ⊕ gyr [ x, u ]( ⊖ u )= b α,m ⊕ gyr [ x, u ]( ⊖ u ) ∈ b α,m ⊕ gyr [ x, u ]( V n +1 )= b α,m ⊕ V n +1 . Therefore, there exists i ≥ i such that x i ∈ b α,m ⊕ V n +1 whenever i ≥ i , whence { x } ∪ { x i : i ≥ i } ⊂ π − ( P α ) ∩ ( b α,m ⊕ V n +1 ).Therefore, we conclude that G has a star-countable cs -network. (cid:3) TRONGLY TOPOLOGICAL GYROGROUPS AND QUOTIENT WITH RESPECT TO L -SUBGYROGROUPS9 Theorem 3.9.
Let G be a strongly topological gyrogroup with a symmetric neighborhoodbase U at and let H be a second-countable admissible subgyrogroup generated from U . If the quotient space G/H has a star-countable wcs ∗ -network, G has also a star-countable wcs ∗ -network.Proof. Let U be a symmetric neighborhood base at 0 such that gyr [ x, y ]( U ) = U forany x, y ∈ G and U ∈ U . Since the L -subgyrogroup H of G is first-countable at theidentity element 0 of G , there exists a countable family { V n : n ∈ N } of open symmetricneighborhoods of 0 in G such that ( V n +1 ⊕ ( V n +1 ⊕ V n +1 )) ⊂ V n for each n ∈ N and thefamily { V n ∩ H : n ∈ N } is a local base at 0 for H .We construct P and F as the same with in Theorem 3.8, and we show that F is a wcs ∗ -network for G .Let { x i } i be a sequence converging to a point x in G and U be a neighborhood of x in G . Choose an open neighborhood V of 0 in G such that ( x ⊕ ( V ⊕ V )) ⊂ U . Since { V n ∩ H : n ∈ N } is a local base at 0 for H , there exists n ∈ N such that V n ∩ H ⊂ V ∩ H .Since P is a wcs ∗ -network for G/H , there exists a subsequence { π ( x i j ) } j of the sequence { π ( x i ) } i such that { π ( x i j ) : j ∈ N } ⊂ P α ⊂ π (( x ⊕ V n +1 ) ∩ ( x ⊕ V )) for some α ∈ Λ.As the sequence { x i } i converges to x , we have some x i j ∈ x ⊕ V n +2 for each j ∈ N .Furthermore, since x i ∈ π − ( P α ), ( x i ⊕ V n +2 ) ∩ π − ( P α ) is non-empty and open in π − ( P α ). Then it follows from B α = π − ( P α ) that there exists m ∈ N such that b α,m ∈ x i ⊕ V n +2 . Then b α,m ∈ x i ⊕ V n +2 ⊂ ( x ⊕ V n +2 ) ⊕ V n +2 = x ⊕ ( V n +2 ⊕ gyr [ V n +2 , x ]( V n +2 ))= x ⊕ ( V n +2 ⊕ V n +2 ) . Moreover, it is proved in Theorem 3.8 that π − ( P α ) ∩ ( b α,m ⊕ V n +1 ) ⊂ U .In conclusion, G has a star-countable wcs ∗ -network. (cid:3) Lemma 3.10. [24]
The following are equivalent for a rectifiable space.(i) Every compact (countably compact) subset is first-countable.(ii) Every compact (countably compact) subset is metrizable.
Theorem 3.11.
Let H be an L -subgyrogroup of a topological gyrogroup G , and supposethat all compact subspaces of H and G/H are metrizable. Then all compact subspacesof G are metrizable as well.Proof. Let π be the natural homomorphism from G onto its quotient space G/H of leftcosets. For an arbitrary y ∈ G/H , there exists a point x ∈ G such that π ( x ) = y . Then π − ( y ) = x ⊕ H which is homeomorphic to H .Fix a compact subset X of G , let f be the restriction of π to X . The compactsubspace Y = f ( X ) of the space G/H is metrizable. Indeed, all compact subsets ofthe fibers of f are metrizable. Since X is compact and f : X → Y is continuous, itis clear that f is closed mapping. By [1, Lemma 3.3.23], all compact subsets of G arefirst-countable. Finally, it follows from Lemma 3.10 that X is metrizable. (cid:3) Lemma 3.12. [31, Lemma 2.1.6]
Let P be a point-countable family of subsets of a space X . Then P is a k -network for X if and only if it is a wcs ∗ -network for X and eachcompact subset of X is first-countable (or sequential). Theorem 3.13.
Let G be a strongly topological gyrogroup with a symmetric neighbor-hood base U at and let H be a second-countable admissible subgyrogroup generated from U . If the quotient space G/H has a star-countable k -network, G has also a star-countable k -network.Proof. Since
G/H has a star-countable k -network, it follows from Theorem 3.9 that G has a star-countable wcs ∗ -network. By Lemma 3.12, each compact subset of G/H isfirst-countable. Then every compact subset of G is first-countable by [1, Lemma 3.3.23]and Theorem 3.11. Therefore, G has a star-countable k -network by Lemma 3.12. (cid:3) Quotient with respect to locally compact admissible L -subgyrogroups In this section, we research the quotient space
G/H with some generalized metricproperties when G is a strongly topological gyrogroup with a symmetric neighborhoodbase U at 0 and H is a locally compact metrizable admissible subgyrogroup generatedfrom U . Suppose that G is a strongly topological gyrogroup with a symmetric neigh-borhood base U at 0 and H is a locally compact metrizable admissible subgyrogroupgenerated from U . We show that if the quotient space G/H is sequential, then G isalso sequential; If the quotient space G/H is strictly (strongly) Fr´echet-Urysohn, then G is also strictly (strongly) Fr´echet-Urysohn; Finally, if the quotient space G/H is astratifiable space (semi-stratifiable space, σ -space, k -semistratifiable space), then G isa local stratifiable space (semi-stratifiable space, σ -space, k -semistratifiable space).First, recall some concepts about convergence and the relations among them. Definition 4.1. [17] Let X be a topological space. A subset A of X is called sequentiallyclosed if no sequence of points of A converges to a point not in A . A subset A of X is called sequentially open if X \ A is sequentially closed. X is called sequential if eachsequentially closed subset of X is closed. Definition 4.2. [17] Let X be a topological space. A space is called Fr´echet-Urysohnat a point x ∈ X if for every A ⊂ X with x ∈ A ⊂ X there is a sequence { x n } n in A such that { x n } n converges to x in X . A space is called Fr´echet-Urysohn if it isFr´echet-Urysohn at every point x ∈ X . Definition 4.3. [20]([35]) Let X be a topological space. A space is called strictly(strongly) Fr´echet-Urysohn at a point x ∈ X if whenever { A n } n is a sequence (decreasingsequence) of subsets in X and x ∈ T n ∈ N A n , there exists x n ∈ A n for each n ∈ N suchthat the sequence { x n } n converges to x . A space X is called strictly (strongly) Fr´echet-Urysohn if it is strictly (strongly) Fr´echet-Urysohn at every point x ∈ X .It is well-known [34] that(1) every first-countable space is a strictly Fr´echet-Urysohn space;(2) every strictly Fr´echet-Urysohn space is a strongly Fr´echet-Urysohn space;(3) every strongly Fr´echet-Urysohn space is a Fr´echet-Urysohn space;(4) every Fr´echet-Urysohn space is a sequential space. Lemma 4.4. [7]
Suppose that ( G, τ, ⊕ ) is a strongly topological gyrogroup with a sym-metric neighborhood base U at , and suppose that H is a locally compact admissiblesubgyrogroup generated from U . Then there exists an open neighborhood U of the iden-tity element such that π ( U ) is closed in G/H and the restriction of π to U is a perfectmapping from U onto the subspace π ( U ) , where π : G → G/H is the natural quotientmapping from G onto the quotient space G/H . TRONGLY TOPOLOGICAL GYROGROUPS AND QUOTIENT WITH RESPECT TO L -SUBGYROGROUPS11 Theorem 4.5.
Suppose that G is a strongly topological gyrogroup with a symmetricneighborhood base U at . Suppose further that H is a locally compact metrizable ad-missible subgyrogroup generated from U such that the quotient space G/H is sequential,then G is also sequential.Proof. By the hypothesis, we assume that G is a strongly topological gyrogroup with asymmetric neighborhood base U at 0. It follows from Lemma 4.4 that there is an openneighborhood U of the identity element 0 in G such that π | U : U → π ( U ) is a perfectmapping and π ( U ) is closed in G/H . Claim 1.
Assume that { x n } n is a sequence in U such that { π ( x n ) } n is a convergentsequence in π ( U ). If x is an accumulation point of the sequence { x n } n , then there is asubsequence of { x n } n which converges to x .Since π | U is perfect, every subsequence of { x n } n has an accumulation point in U .Put F = π − ( π ( x )) ∩ U . By the assumption, π − ( π ( x )) = x ⊕ H is metrizable. Sinceevery topological gyrogroup is regular, there exists a sequence { U k } k of open subsets in G such that U k +1 ⊂ U k for each k ∈ N and { x } = F ∩ T k ∈ N U k . Choose a subsequence { x n k } k of { x n } n such that x n k ∈ U k for each k ∈ N . For an arbitrary accumulationpoint p of a subsequence of the sequence { x n k } k , we have π ( p ) = π ( x ) and p ∈ T k ∈ N U k .Thus p = x . Therefore, x is the unique accumulation point of every subsequence of { x n k } k , proving that x n k → x .Choose an open neighborhood V of 0 such that V ⊂ U . Claim 2. If C is sequentially closed in V , then π ( C ) is closed in π ( V ).Suppose that { y n } n is a sequence in π ( C ) such that y n → y in π ( V ). Choose x n ∈ C with π ( x n ) = y n for each n ∈ N . Since every subsequence of the sequence { x n } n hasan accumulation point, it follows from Claim 1 that there exist a point x ∈ π − ( y ) anda subsequence { x n k } k of { x n } n such that x n k → x . Since C is sequentially closed, weobtain x ∈ C and y ∈ π ( C ). Therefore, π ( C ) is sequentially closed in π ( V ). Since π | U : U → π ( U ) is a closed mapping and π ( U ) is closed in G/H , π ( V ) is closed in G/H .Since
G/H is sequential, π ( V ) is also sequential and then π ( C ) is closed in π ( V ). Claim 3. V is a sequential subspace.Suppose on the contrary, there is a non-closed and sequentially closed subset A of V .Then there exists a point x such that x ∈ cl V ( A ) \ A . It is clear that cl V ( A ) = A . Let f = π | V : V → π ( V ) and B = A ∩ f − ( f ( x )). Since B is a closed subset of A , B issequentially closed. Moreover, the fiber f − ( f ( x )) = ( π − ( π ( x ))) ∩ V is sequential, so B is closed in V . Since x B , there exists an open neighborhood W of x in V suchthat W ∩ B = ∅ . Let C = W ∩ A , then C is also sequentially closed as a closed subsetof A and x ∈ C \ C . Therefore, C ∩ f − ( f ( x )) = W ∩ B = ∅ , then f ( x ) ∈ f ( C ) \ f ( C ).So f ( C ) = π ( C ) is not closed in π ( V ) which is contradict with Claim 2.Since G is homogeneous and by Claim 3, we obtain that G is a locally sequentialspace. Hence, G is sequential space. (cid:3) Lemma 4.6. [1, Proposition 4.7.18]
Suppose that X is a regular space, and that f : X → Y is a closed mapping. Suppose also that b ∈ X is a G δ -point in the space F = f − ( f ( b )) (i.e., the singleton { b } is a G δ -set in the space F ) and F is Fr´echet-Urysohn at b . If the space Y is strongly Fr´echet-Urysohn, then X is Fr´echet-Urysohnat b . Theorem 4.7.
Suppose that G is a strongly topological gyrogroup with a symmetricneighborhood base U at . Suppose further that H is a locally compact metrizable admissible subgyrogroup generated from U such that the quotient space G/H is stronglyFr´echet-Urysohn. Then the space G is also strongly Fr´echet-Urysohn.Proof. Suppose that G is a strongly topological gyrogroup with a symmetric neighbor-hood base U at 0. It follows from Lemma 4.4 that there is an open neighborhood U ofthe identity element 0 in G such that π | U : U → π ( U ) is a perfect mapping and π ( U ) isclosed in G/H .Put f = π | U : U → π ( U ). Then f ( U ) = π ( U ) is strongly Fr´echet-Urysohn. Foreach b ∈ U , f − ( f ( b )) = π − ( π ( b )) ∩ U = ( b ⊕ H ) ∩ U is metrizable. Therefore, thesingleton { b } is a G δ -set in the space f − ( f ( b )). Moreover, since the quotient space G/H is strongly Fr´echet-Urysohn, the space G is locally Fr´echet-Urysohn by Lemma4.6. Hence, G is Fr´echet-Urysohn. Furthermore, every Fr´echet-Urysohn topologicalgyrogroup is strongly Fr´echet-Urysohn by [24, Corollary 5.2]. So G is strongly Fr´echet-Urysohn. (cid:3) Lemma 4.8. [29]
Suppose that X is a regular space, and that f : X → Y is a closedmapping. Suppose also that b ∈ X is a G δ -point in the space F = f − ( f ( b )) (i.e.,the singleton { b } is a G δ -set in the space F ) and F is countably compact and strictlyFr´echet-Urysohn at b . If the space Y is strictly Fr´echet-Urysohn at f ( b ) , then X isstrictly Fr´echet-Urysohn at b . Theorem 4.9.
Suppose that G is a strongly topological gyrogroup with a symmetricneighborhood base U at . Suppose further that H is a locally compact metrizableadmissible subgyrogroup generated from U such that the quotient space G/H is strictlyFr´echet-Urysohn, then G is also strictly Fr´echet-Urysohn.Proof. By the hypothesis, we assume that G is a strongly topological gyrogroup with asymmetric neighborhood base U at 0. It follows from Lemma 4.4 that there is an openneighborhood U of the identity element 0 in G such that π | U : U → π ( U ) is a perfectmapping and π ( U ) is closed in G/H .Put f = π | U : U → π ( U ). Then f ( U ) = π ( U ) is strictly Fr´echet-Urysohn. For each b ∈ U , f − ( f ( b )) = π − ( π ( b )) ∩ U = ( b ⊕ H ) ∩ U is compact and metrizable. It followsfrom Lemma 4.8 that U is strictly Fr´echet-Urysohn. Therefore, G is locally strictlyFr´echet-Urysohn and G is strictly Fr´echet-Urysohn. (cid:3) Definition 4.10. [11, 32] A topological space (
X, τ ) is semi-stratifiable if there is afunction S : N × τ → { closed subsets of X } such that(a) if U ∈ τ , then U = S ∞ n =1 S ( n, U );(b) if U, V ∈ τ and U ⊂ V , then S ( n, U ) ⊂ S ( n, V ) for each n ∈ N .The function S is called a semistratification of X . (If, in addition, the function S satisfies U = S ∞ n =1 [ S ( n, U )] ◦ for each U ∈ τ , then S is called a stratification of X and X is said to be stratifiable [8].)The concept of k -semistratifiable space was introduced in [23]. Theorem 4.11.
Let G be a strongly topological gyrogroup with a symmetric neighbor-hood base U at . Suppose that H is a locally compact metrizable admissible subgy-rogroup generated from U such that the quotient space G/H has property P , where P is a topological property. Then the space G is locally in P if P satisfies the following:(1) P is closed hereditary;(2) P contains point G δ -property, and TRONGLY TOPOLOGICAL GYROGROUPS AND QUOTIENT WITH RESPECT TO L -SUBGYROGROUPS13 (3) let f : X → Y be a perfect mapping, if X has G δ -diagonal and Y is P , then X is P .Proof. Suppose that π : G → G/H is the canonical homomorphism. Since
G/H is in P and P contains point G δ -property, { H } is a G δ -subset in G/H , that is, there existsa sequence { V n : n ∈ N } of open sets in G/H such that { H } = T n ∈ N V n . Therefore, H = T n ∈ N π − ( V n ). Since H is a metrizable L -subgyrogroup of G , there is a family { W n : n ∈ N } of open neighborhoods of the identity element 0 such that { W n ∩ H : n ∈ N } is an open countable neighborhood base in H . Hence, { } = \ n ∈ N ( W n ∩ H ) = \ n ∈ N ( W n ∩ π − ( V n )) . Then G has point G δ -property. It follows from [6] that every strongly topologicalgyrogroup with countable pseudocharacter is submetrizable. So G has G δ -diagonal.By Lemma 4.4, there is an open neighborhood U of the identity element 0 in G suchthat π | U : U → π ( U ) is a perfect mapping and π ( U ) is closed in G/H . Then by (1) and(3), the subspace U is in P . Therefore, G is locally in P . (cid:3) Note that every stratifiable space, semi-stratifiable space and σ -space satisfies theconditions in Theorem 4.11, respectively. Corollary 4.12.
Suppose that G is a strongly topological gyrogroup with a symmet-ric neighborhood base U at . Suppose further that H is a locally compact metriz-able admissible subgyrogroup generated from U such that the quotient space G/H is astratifiable space (semi-stratifiable space, σ -space). Then G is a local stratifiable space(semi-stratifiable space, σ -space). Definition 4.13. [28] Suppose that { V n } is a sequence of open covers of a space.(1) { V n } is said to be a G δ - diagonal sequence for X if { x } = T n ∈ N st ( x, V n ) for each x ∈ X .(2) { V n } is said to be a KG - sequence for X if x n ∈ st ( a n , V n ) for each n ∈ N , and x n → p , a n → q , then p = q .It was claimed in [28] that if f : X → Y is a perfect map and Y is a k -semistratifiablespace, then X is a k -semistratifiable space if and only if X has a KG -sequence. Theorem 4.14.
Let G be a strongly topological gyrogroup. If G has point G δ -property, G has a KG -sequence.Proof. Suppose that G is a strongly topological gyrogroup with a symmetric neighbor-hood base U at 0. Since G has point G δ -property, there exists a sequence { V n } n ofopen neighborhoods of the identity element 0 such that T n ∈ N V n = { } . For each V n ,there exists U n ∈ U such that U n ⊕ U n ⊂ V n . Put U n = { x ⊕ U n : x ∈ G } . It is clearthat each U n is an open cover of G . Claim. {U n } n ∈ N is a KG -sequence in G .Let p n ∈ st ( q n , U n ), where { q n } → q and p n → p . For each n ∈ N , we can find x n ∈ G such that p n , q n ∈ ( x n ⊕ U n ). Then, there are v n , u n ∈ U n such that p n = x n ⊕ u n and q n = x n ⊕ v n . Therefore, by Lemma 2.2, x n = ( x n ⊕ v n ) ⊕ gyr [ x n , v n ]( ⊖ v n )= q n ⊕ gyr [ x n , v n ]( ⊖ v n ) ∈ q n ⊕ gyr [ x n , v n ]( U n )= q n ⊕ U n . Then, p n = x n ⊕ u n ∈ ( q n ⊕ U n ) ⊕ u n = q n ⊕ ( U n ⊕ gyr [ U n , q n ]( u n )) ⊂ q n ⊕ ( U n ⊕ gyr [ U n , q n ]( U n ))= q n ⊕ ( U n ⊕ U n ) ⊂ q n ⊕ V n . Therefore, ⊖ q n ⊕ p n ∈ V n for each n ∈ N . Hence, ⊖ q n ⊕ p n ∈ T n ∈ N V n = { } , that is, p = q .We conclude that G has a KG -sequence. (cid:3) Naturally, we have the following result.
Corollary 4.15.
Suppose that G is a strongly topological gyrogroup with a symmet-ric neighborhood base U at . Suppose further that H is a locally compact metriz-able admissible subgyrogroup generated from U such that the quotient space G/H is k -semistratifiable, then the space G is locally k -semistratifiable. Finally, we pose the following questions.
Question 4.16.
Let P be any calss of topological spaces which is closed hereditary andclosed under locally finite unions of closed sets. Is every strongly topological gyrogroupwhich is locally in P in P ? Clearly, if the question is affirmative, the result G is a local stratifiable space (semi-stratifiable space, σ -space) in Corollary 4.12 will be strengthened directly. Question 4.17.
Let G be a strongly topological gyrogroup with a symmetric neighbor-hood base U at and let H be an admissible subgyrogroup generated from U . Is thequotient space G/H completely regular?
Acknowledgements . The first author would like to express his congratulationsto his supervisor Professor Xiaoquan Xu on the occasion of his 60th birthday. Theauthors are thankful to the anonymous referees for valuable remarks and correctionsand all other sort of help related to the content of this article.
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Embedding of locally compact Hausdorff topologicalgyrogroups in topological groups , Topol. Appl. 273(2020), Article 107102. (Meng Bao): College of Mathematics, Sichuan University, Chengdu 610064, P. R. China
Email address : [email protected] (Xuewei Ling): Institute of Mathematics, Nanjing Normal University, Nanjing, Jiangsu210046, P.R. China Email address : (Xiaoquan Xu): School of mathematics and statistics, Minnan Normal University, Zhangzhou363000, P. R. China Email address ::