# Quotient spaces with strong subgyrogroups 111This research was supported by
the National Natural Science Foundation of China (Nos. 12071199, 11661057),
the Natural Science Foundation of Jiangxi Province, China (No.
20192ACBL20045).

Meng Bao<EMAIL_ADDRESS>College of Mathematics, Sichuan University,
Chengdu 610064, China  Xuewei Ling<EMAIL_ADDRESS>Institute of Mathematics,
Nanjing Normal University, Nanjing 210046, China  Xiaoquan Xu
<EMAIL_ADDRESS>Fujian Key Laboratory of Granular Computing and
Applications, Minnan Normal University, Zhangzhou 363000, China

###### Abstract

In this paper, we mainly investigate the quotient spaces $G/H$ when $G$ is a
strongly topological gyrogroup and $H$ is a strong subgyrogroup of $G$. It is
shown that if $G$ is a strongly topological gyrogroup, $H$ is a closed strong
subgyrogroup of $G$ and $H$ is inner neutral, then the quotient space $G/H$ is
first-countable if and only if $G/H$ is a bisequential space if and only if
$G/H$ is a weakly first-countable space if and only if $G/H$ is a
$csf$-countable and sequential $\alpha_{7}$-space. Moreover, it is shown that
if $G$ is a strongly topological gyrogroup and $H$ is a locally compact strong
subgyrogroup of $G$, then there exists an open neighborhood $U$ of the
identity element $0$ such that $\pi(\overline{U})$ is closed in $G/H$ and the
restriction of $\pi$ to $\overline{U}$ is a perfect mapping from
$\overline{U}$ onto the subspace $\pi(\overline{U})$; if $H$ is a locally
compact metrizable strong subgyrogroup of $G$ and the quotient space $G/H$ is
sequential, then $G$ is also sequential; if $H$ is a closed first-countable
and separable strong subgyrogroup of $G$, the quotient space $G/H$ is an
$\aleph_{0}$-space, then $G$ is an $\aleph_{0}$-space; if the quotient space
$G/H$ is a cosmic space, then $G$ is also a cosmic space; if the quotient
space $G/H$ has a star-countable $cs$-network or star-countable
$wcs^{*}$-network, then $G$ also has a star-countable $cs$-network or star-
countable $wcs^{*}$-network, respectively.

###### keywords:

Topological gyrogroups, strong subgyrogroups, metrizable, quotient spaces

###### MSC:

22A22; 54A20; 20N05; 18A32; 20B30.

††journal: Topology and its Applications

##  1 Introduction

Let $c$ be a positive constant representing the speed of light in vacuum and
$\mathbb{R}_{c}^{3}=\\{\mathbf{v}\in\mathbb{R}^{3}:||\mathbf{v}||<c\\}$ the
$c$-ball of relativistically admissible velocities with Einstein velocity
addition. It is well-known that Einstein velocity addition is the standard
velocity addition of relativistically admissible velocities that Einstein
introduced in 1905 that founded the special theory of relativity. The Einstein
velocity addition $\oplus_{E}$ is given as the following:

$\mathbf{u}\oplus_{E}\mathbf{v}=\frac{1}{1+\frac{\mathbf{u}\cdot\mathbf{v}}{c^{2}}}(\mathbf{u}+\frac{1}{\gamma_{\mathbf{u}}}\mathbf{v}+\frac{1}{c^{2}}\frac{\gamma_{\mathbf{u}}}{1+\gamma_{\mathbf{u}}}(\mathbf{u}\cdot\mathbf{v})\mathbf{u}),$

where $\mathbf{u,v}\in\mathbb{R}_{c}^{3}$ and $\gamma_{\mathbf{u}}$ is given
by

$\gamma_{\mathbf{u}}=\frac{1}{\sqrt{1-\frac{\mathbf{u}\cdot\mathbf{u}}{c^{2}}}}.$

By the study of the $c$-ball of relativistically admissible velocities, A.A.
Ungar discovered that the seemingly structureless Einstein addition of
relativistically admissible velocities possesses a rich grouplike structure
and he posed the concept of gyrogroups in [60]. Indeed, a gyrogroup is the
most natural extension of a group into the regime of the nonassociative
algebra that we need for extending analytic Euclidean geometry into analytic
hyperbolic geometry. In 2017, the gyrogroup was equipped with a topology by W.
Atiponrat [4] such that the binary operation $\oplus:G\times G\rightarrow G$
is jointly continuous and the inverse mapping $\ominus(\cdot):G\rightarrow G$,
i.e. $x\rightarrow\ominus x$, is also continuous and she called it topological
gyrogroups. At the same time, she asked whether the first-countability axiom
can imply metrizability in a Hausdorff topological gyrogroup? Shortly
afterwards, Cai, Lin and He in [19] proved that every topological gyrogroup is
a rectifiable space, which implies that the equivalence between the axioms of
first-countable and metrizability in a Hausdorff topological gyrogroup. By
further researches of the classical Möbius gyrogroups, Einstein gyrogroups,
and Proper Velocity gyrogroups, Bao and Lin [8] found that each of them has an
open neighborhood base at the identity element $0$ such that all elements of
the base are invariant under the groupoid automorphisms with standard
topology. Therefore, they posed the concept of strongly topological gyrogroups
and showed that every feathered strongly topological gyrogroup is paracompact.
A series of results on topological gyrogroups and strongly topological
gyrogroups have been obtained in [3, 5, 6, 9, 10, 11, 12, 13, 15, 16, 36, 37,
38]. In particular, it was proved in [9, 10] that each $T_{0}$-strongly
topological gyrogroup is completely regular, every $T_{0}$-strongly
topological gyrogroup with a countable pseudocharacter is submetrizable and
each locally paracompact strongly topological gyrogroup is paracompact.

In [54], T. Suksumran and K. Wiboonton posed the notion of $L$-subgyrogroups
and showed that if $H$ is an $L$-subgyrogroup of a gyrogroup $G$, then the set
$\\{a\oplus H:a\in G\\}$ forms a disjoint partition of $G$. Therefore, it is
natural to research the quotient spaces of a strongly topological gyrogroup
with respect to $L$-subgyrogroups as left cosets. Moreover, Bao and Lin proved
that if $G$ is a $T_{0}$-strongly topological gyrogroup with a symmetric
neighborhood base $\mathscr{U}$ at $0$ and $H$ is an admissible subgyrogroup
generated from $\mathscr{U}$, then the left coset space $G/H$ is
submetrizable. Recently, Bao and Xu [14] constructed a subgyrogroup $H$ in a
strongly topological gyrogroup $G$ such that $gyr[x,y](H)=H$ for all $x,y\in
G$, hence they introduced the concept of strong subgyrogroups in a strongly
topological gyrogroup. They showed that if $G$ is a strongly topological
gyrogroup, $H$ is a closed strong subgyrogroup of $G$ and $H$ is inner
neutral, then $G/H$ is first-countable if and only if $G/H$ is Fréchet-Urysohn
with an $\omega^{\omega}$-base.

In this paper, we continue to research the quotient spaces of strongly
topological gyrogroups with respect to closed strong subgyrogroups. First, it
is not difficult to see that if $G$ is a strongly topological gyrogroup with a
symmetric neighborhood base $\mathscr{U}$ at $0$, $P$ is an admissible
subgyrogroup generated from $\mathscr{U}$ and the quotient space $G/P$ is
first-countable, then it is metrizable. However, the two arrows space is a
compact coset space which is first-countable, but not submetrizable, and it is
homeomorphic to $G/H$, where $G$ is a strongly topological gyrogroup and $H$
is a closed strong subgyrogroup of $G$. Moreover, it is shown that if $G$ is a
strongly topological gyrogroup, $H$ is a closed strong subgyrogroup of $G$ and
$H$ is inner neutral, then the quotient space $G/H$ is first-countable if and
only if $G/H$ is a bisequential space if and only if $G/H$ is a weakly first-
countable space if and only if $G/H$ is a $csf$-countable and sequential
$\alpha_{7}$-space. More important, it is shown that if $G$ is a strongly
topological gyrogroup and $H$ is a locally compact strong subgyrogroup of $G$,
then there exists an open neighborhood $U$ of the identity element $0$ such
that $\pi(\overline{U})$ is closed in $G/H$ and the restriction of $\pi$ to
$\overline{U}$ is a perfect mapping from $\overline{U}$ onto the subspace
$\pi(\overline{U})$, which implies that if $H$ is a locally compact strong
subgyrogroup of a strongly topological gyrogroup $G$ and the quotient space
$G/H$ has some nice properties, such as locally compact, locally countably
compact, locally pseudocompact, etc., then $G$ also has the same properties.
Then, we show that when $G$ is a strongly topological gyrogroup and $H$ is a
locally compact metrizable strong subgyrogroup of $G$, if the quotient space
$G/H$ is strictly (strongly) Fréchet-Urysohn, then $G$ is also strictly
(strongly) Fréchet-Urysohn; if the quotient space $G/H$ is sequential, then
$G$ is also sequential. Therefore, some important and interesting results in
[44] are improved. Finally, we study the quotient space $G/H$ with some
generalized metric properties, where $G$ is a strongly topological gyrogroup
and $H$ is a closed first-countable and separable strong subgyrogroup of $G$.
We prove that if the quotient space $G/H$ is an $\aleph_{0}$-space, then $G$
is an $\aleph_{0}$-space; if the quotient space $G/H$ is a cosmic space, then
$G$ is also a cosmic space; if the quotient space $G/H$ has a star-countable
$cs$-network or star-countable $wcs^{*}$-network, then $G$ also has a star-
countable $cs$-network or star-countable $wcs^{*}$-network, respectively.

##  2 Preliminary

Throughout this paper, all topological spaces are assumed to be Hausdorff,
unless otherwise is explicitly stated. Let $\mathbb{N}$ be the set of all
positive integers and $\omega$ the first infinite ordinal. The readers may
consult [2, 23, 45, 58] for notation and terminology not explicitly given
here. Next we recall some definitions and facts.

######  Definition 2.1.

[4] Let $G$ be a nonempty set, and let $\oplus:G\times G\rightarrow G$ be a
binary operation on $G$. Then the pair $(G,\oplus)$ is called a groupoid. A
function $f$ from a groupoid $(G_{1},\oplus_{1})$ to a groupoid
$(G_{2},\oplus_{2})$ is called a groupoid homomorphism if
$f(x\oplus_{1}y)=f(x)\oplus_{2}f(y)$ for any elements $x,y\in G_{1}$.
Furthermore, a bijective groupoid homomorphism from a groupoid $(G,\oplus)$ to
itself will be called a groupoid automorphism. We write $\mbox{Aut}(G,\oplus)$
for the set of all automorphisms of a groupoid $(G,\oplus)$.

######  Definition 2.2.

[58] Let $(G,\oplus)$ be a groupoid. The system $(G,\oplus)$ is called a
gyrogroup, if its binary operation satisfies the following conditions:

$(G1)$ There exists a unique identity element $0\in G$ such that $0\oplus
a=a=a\oplus 0$ for all $a\in G$.

$(G2)$ For each $x\in G$, there exists a unique inverse element $\ominus x\in
G$ such that $\ominus x\oplus x=0=x\oplus(\ominus x)$.

$(G3)$ For all $x,y\in G$, there exists
$\mbox{gyr}[x,y]\in\mbox{Aut}(G,\oplus)$ with the property that
$x\oplus(y\oplus z)=(x\oplus y)\oplus\mbox{gyr}[x,y](z)$ for all $z\in G$.

$(G4)$ For any $x,y\in G$, $\mbox{gyr}[x\oplus y,y]=\mbox{gyr}[x,y]$.

######  Lemma 2.3.

[58] Let $(G,\oplus)$ be a gyrogroup. Then for any $x,y,z\in G$, we obtain the
followings:

  1. 1.

$(\ominus x)\oplus(x\oplus y)=y$. (left cancellation law)

  2. 2.

$(x\oplus(\ominus y))\oplus\mbox{gyr}[x,\ominus y](y)=x$. (right cancellation
law)

  3. 3.

$(x\oplus\mbox{gyr}[x,y](\ominus y))\oplus y=x$.

  4. 4.

$\mbox{gyr}[x,y](z)=\ominus(x\oplus y)\oplus(x\oplus(y\oplus z))$.

  5. 5.

$(x\oplus y)\oplus z=x\oplus(y\oplus\mbox{gyr}[y,x](z))$.

######  Proposition 2.4.

[54] Let $G$ be a gyrogroup and let $X\subseteq G$. Then the followings are
equivalent:

(1) $\mbox{gyr}[a,b](X)\subseteq X$ for all $a,b\in G$;

(2) $\mbox{gyr}[a,b](X)=X$ for all $a,b\in G$.

Notice that a group is a gyrogroup $(G,\oplus)$ such that $\mbox{gyr}[x,y]$ is
the identity function for all $x,y\in G$. The definition of a subgyrogroup is
as follows.

######  Definition 2.5.

[54] Let $(G,\oplus)$ be a gyrogroup. A nonempty subset $H$ of $G$ is called a
subgyrogroup, denoted by $H\leq G$, if $H$ forms a gyrogroup under the
operation inherited from $G$ and the restriction of $\mbox{gyr}[a,b]$ to $H$
is an automorphism of $H$ for all $a,b\in H$.

Furthermore, a subgyrogroup $H$ of $G$ is said to be an $L$-subgyrogroup,
denoted by $H\leq_{L}G$, if $\mbox{gyr}[a,h](H)=H$ for all $a\in G$ and $h\in
H$.

######  Definition 2.6.

[4] A triple $(G,\tau,\oplus)$ is called a topological gyrogroup if the
following statements hold:

(1) $(G,\tau)$ is a topological space.

(2) $(G,\oplus)$ is a gyrogroup.

(3) The binary operation $\oplus:G\times G\rightarrow G$ is jointly continuous
while $G\times G$ is endowed with the product topology, and the operation of
taking the inverse $\ominus(\cdot):G\rightarrow G$, i.e. $x\rightarrow\ominus
x$, is also continuous.

Obviously, every topological group is a topological gyrogroup. However, every
topological gyrogroup whose gyrations are not identically equal to the
identity is not a topological group. In particular, it was proved in [4] that
the Einstein gyrogroup with the standard topology is a topological gyrogroup
but not a topological group. Next, we introduce the definition of a strongly
topological gyrogroup, it is very important in this paper.

######  Definition 2.7.

([8]) Let $G$ be a topological gyrogroup. We say that $G$ is a strongly
topological gyrogroup if there exists a neighborhood base $\mathscr{U}$ of $0$
such that, for every $U\in\mathscr{U}$, $\mbox{gyr}[x,y](U)=U$ for any $x,y\in
G$. For convenience, we say that $G$ is a strongly topological gyrogroup with
neighborhood base $\mathscr{U}$ of $0$.

For each $U\in\mathscr{U}$, we can set $V=U\cup(\ominus U)$. Then,

$\mbox{gyr}[x,y](V)=\mbox{gyr}[x,y](U\cup(\ominus
U))=\mbox{gyr}[x,y](U)\cup(\ominus\mbox{gyr}[x,y](U))=U\cup(\ominus U)=V,$

for all $x,y\in G$. Obviously, the family $\\{U\cup(\ominus
U):U\in\mathscr{U}\\}$ is also a neighborhood base of $0$. Therefore, we may
assume that $U$ is symmetric for each $U\in\mathscr{U}$ in Definition 2.7.
Moreover, in the classical Möbius, Einstein, or Proper Velocity gyrogroups, we
know that gyrations are indeed special rotations, however for an arbitrary
gyrogroup, gyrations belong to the automorphism group of $G$ and need not be
necessarily rotations.

In [8], the authors proved that there is a strongly topological gyrogroup
which is not a topological group, see Example 2.8.

######  Example 2.8.

[8] Let $\mathbb{D}$ be the complex open unit disk
$\\{z\in\mathbb{C}:|z|<1\\}$. We consider $\mathbb{D}$ with the standard
topology. In [4, Example 2], define a Möbius addition
$\oplus_{M}:\mathbb{D}\times\mathbb{D}\rightarrow\mathbb{D}$ to be a function
such that

$a\oplus_{M}b=\frac{a+b}{1+\bar{a}b}\ \mbox{for all}\ a,b\in\mathbb{D}.$

Then $(\mathbb{D},\oplus_{M})$ is a gyrogroup, and it follows from [4, Example
2] that

$\mbox{gyr}[a,b](c)=\frac{1+a\bar{b}}{1+\bar{a}b}c\ \mbox{for any}\
a,b,c\in\mathbb{D}.$

For any $n\in\mathbb{N}$, let $U_{n}=\\{x\in\mathbb{D}:|x|\leq\frac{1}{n}\\}$.
Then, $\mathscr{U}=\\{U_{n}:n\in\mathbb{N}\\}$ is a neighborhood base of $0$.
Moreover, we observe that $|\frac{1+a\bar{b}}{1+\bar{a}b}|=1$. Therefore, we
obtain that $\mbox{gyr}[x,y](U)\subseteq U$, for any $x,y\in\mathbb{D}$ and
each $U\in\mathscr{U}$, then it follows that $\mbox{gyr}[x,y](U)=U$ by
Proposition 2.4. Hence, $(\mathbb{D},\oplus_{M})$ is a strongly topological
gyrogroup. However, $(\mathbb{D},\oplus_{M})$ is not a group by [4, Example
2].

Then, Bao and Lin constructed the following example by Example 2.8 to show
that for any cardinality $\kappa>\omega$, there exists a gyrogroup $G$ with
subgyrogroup $H$ of the cardinality $\kappa$ such that $H$ is not a group. The
example guarantees the existence of gyrogroups with cardinality $\kappa$ such
that $\kappa>\omega$, which is important for the research of infinite
gyrogroups.

######  Example 2.9.

[11] Let $\mathbb{D}$ be the gyrogroup in Example 2.8 and let $\kappa$ be an
infinite cardinal number. It follows from [56, Theorem 2.1] that
$\mathbb{D}^{\kappa}$ is a gyrogroup. Fix a subset $X$ of the gyrogroup
$\mathbb{D}^{\kappa}$ such that the cardinality of $X$ is equal to $\kappa$
and $X$ contains arbitrary three points $x=(x_{\alpha})_{\alpha<\kappa}$,
$y=(y_{\alpha})_{\alpha<\kappa}$ and $z=(z_{\alpha})_{\alpha<\kappa}$ of
$\mathbb{D}^{\kappa}$ such that there exists $\beta<\alpha$ with
$x_{\beta}=1/2,y_{\beta}=i/2$ and $z_{\beta}=-1/2$. From the proof of [4,
Example 2], we see that $x\oplus(y\oplus z)\neq(x\oplus y)\oplus z$. Put
$H=\langle X\rangle$, that is, $H$ is a subgyrogroup generated from $X$. Then
the cardinality of $H$ is also equal to $\kappa$. Moreover, since $x,y,z\in
H$, it follows that $H$ is not a group.

We recall the following concept of the coset space of a topological gyrogroup.

Let $(G,\tau,\oplus)$ be a topological gyrogroup and $H$ an $L$-subgyrogroup
of $G$. It follows from [54, Theorem 20] that $G/H=\\{a\oplus H:a\in G\\}$ is
a coset space which defines a partition of $G$. We denote by $\pi$ the mapping
$a\mapsto a\oplus H$ from $G$ onto $G/H$. Clearly, for each $a\in G$, we have
$\pi^{-1}(\pi(a))=a\oplus H$. Indeed, for any $a\in G$ and $h\in H$,

$\displaystyle(a\oplus h)\oplus H$ $\displaystyle=$ $\displaystyle
a\oplus(h\oplus\mbox{gyr}[h,a](H))$ $\displaystyle=$ $\displaystyle
a\oplus(h\oplus\mbox{gyr}^{-1}[a,h](H))$ $\displaystyle=$ $\displaystyle
a\oplus(h\oplus H)$ $\displaystyle=$ $\displaystyle a\oplus H$

Denote by $\tau(G)$ the topology of $G$, the quotient topology on $G/H$ is as
follows:

$\tau(G/H)=\\{O\subseteq G/H:\pi^{-1}(O)\in\tau(G)\\}.$

Throughout this paper, denote by $\pi$ the natural homomorphism from a
topological gyrogroup $G$ to its quotient topology on $G/H$.

Then we recall some important concepts in the following researches.

######  Definition 2.10.

Let $X$ be a topological space.

$(1)$ $X$ is called a weakly first-countable space or $gf$-countable space [1]
if for each point $x\in X$ it is possible to assign a sequence
$\\{B(n,x):n\in\mathbb{N}\\}$ of subsets of $X$ containing $x$ in such a way
that $B(n+1,x)\subseteq B(n,x)$ and so that a set $U$ is open if, and only if,
for each $x\in U$ there exists $n\in\mathbb{N}$ such that $B(n,x)\subseteq U$.

$(2)$ $X$ is called a sequential space [29] if for each non-closed subset
$A\subseteq X$, there are a point $x\in X\setminus A$ and a sequence in $A$
converging to $x$ in $X$.

$(3)$ $X$ is called a Fréchet-Urysohn space [29] if for any subset $A\subseteq
X$ and $x\in\overline{A}$, there is a sequence in $A$ converging to $x$ in
$X$.

$(4)$ $X$ is called a strongly Fréchet-Urysohn space [52] if the following
condition is satisfied:

(SFU) For each $x\in X$ and every sequence $\xi=\\{A_{n}:n\in\mathbb{N}\\}$ of
subsets of $X$ such that $x\in\bigcap_{n\in\mathbb{N}}\overline{A_{n}}$, there
exists a sequence $\eta=\\{b_{n}:n\in\mathbb{N}\\}$ in $X$ converging to $x$
and intersecting infinitely many members of $\xi$.

$(5)$ $X$ is called an $\alpha_{4}$-space [51], if for every point $x\in X$
and each sheaf $\\{S_{n}:n\in\mathbb{N}\\}$ with the vertex $x$, there exists
a sequence converging to $x$ which meets infinitely many sequences $S_{n}$.

$(6)$ $X$ is called an $\alpha_{7}$-space [7], if for every point $x\in X$ and
each sheaf $\\{S_{n}:n\in\omega\\}$ with the vertex $x$, there exists a
sequence converging to some point $y\in X$ which meets infinitely many
sequences $S_{n}$.

######  Definition 2.11.

[2] Let $\zeta$ be a family of non-empty subsets of a topological space $X$.

$(1)$ $\zeta$ is called a prefilter on $X$ if whenever $P_{1}$ and $P_{2}$ are
in $\zeta$, there exists $P\in\zeta$ such that $P\subseteq P_{1}\cap P_{2}$.

$(2)$ A prefilter $\zeta$ on $X$ is said to converge to a point $x\in X$ if
every open neighbourhood of $x$ contains an element of $\zeta$.

$(3)$ A prefilter $\zeta$ on $X$ is said to accumulate to a point $x\in X$ if
$x$ belongs to the closure of each element of $\zeta$.

$(4)$ Two prefilters $\zeta$ and $\eta$ on $X$ are said to be synchronous if,
for any $P\in\eta$ and $Q\in\eta$, $P\cap Q\neq\emptyset$.

$(5)$ $X$ is called a bisequential space if, for every prefilter $\zeta$ on
$X$ accumulating to a point $x\in X$, there exists a countable prefilter
$\eta$ on $X$ converging to the same point $x$ such that $\zeta$ and $\eta$
are synchronous.

The following figure lists some basic relationships of a class of weakly first
countable spaces, see [47].

bisequentialspacestrongly Fréchet-UrysohnspaceFréchet-Urysohnspacefirst-
countablespace$\alpha_{4}$-space$\alpha_{7}$-spacesequentialspaceweakly first-
countablespaceThe relationships among a class of weakly first-countable spaces

##  3 Quotient spaces with inner neutral strong subgyrogroups

In this section, it is shown that if $G$ is a strongly topological gyrogroup,
$H$ is a closed strong subgyrogroup of $G$ and $H$ is inner neutral, then the
quotient space $G/H$ is first-countable if and only if $G/H$ is a bisequential
space if and only if $G/H$ is a weakly first-countable space if and only if
$G/H$ is a $csf$-countable and sequential $\alpha_{7}$-space.

######  Definition 3.1.

[14] A subgyrogroup $H$ of a topological gyrogroup $G$ is called strong
subgyrogroup if for any $x,y\in G$, we have $gyr[x,y](H)=H$.

######  Definition 3.2.

[9] A subgyrogroup $H$ of a topological gyrogroup $G$ is called admissible if
there exists a sequence $\\{U_{n}:n\in\mathbb{N}\\}$ of open symmetric
neighborhoods of the identity $0$ in $G$ such that
$U_{n+1}\oplus(U_{n+1}\oplus U_{n+1})\subseteq U_{n}$ for each
$n\in\mathbb{N}$ and $H=\bigcap_{n\in\mathbb{N}}U_{n}$. If $G$ is a strongly
topological gyrogroup with a symmetric neighborhood base $\mathscr{U}$ at $0$
and each $U_{n}\in\mathscr{U}$, we say that the admissible topological
subgyrogroup is generated from $\mathscr{U}$.

Obviously, every strong subgyrogroup is an $L$-subgyrogroup. Moreover, in a
strongly topological gyrogroup with neighborhood base $\mathscr{U}$ of $0$, it
is not difficult to see that each admissible subgyrogroup generated from
$\mathscr{U}$ is a strong subgyrogroup. Moreover, the authors claimed that
every strongly topological gyrogroup $G$ contains some strong subgyrogroups
which are union-generated from open neighborhoods of the identity element by
construction, see [14, Proposition 3.11].

######  Lemma 3.3.

[8] Let $G$ be a topological gyrogroup and $H$ an $L$-subgyrogroup of $G$.
Then the natural homomorphism $\pi$ from a topological gyrogroup $G$ to its
quotient topology on $G/H$ is an open and continuous mapping.

######  Lemma 3.4.

[14] Let $G$ be a strongly topological gyrogroup and $H$ a closed strong
subgyrogroup of $G$. Then the family $\\{\pi(x\oplus V):V\in\tau,0\in U\\}$ is
a local base of the space $G/H$ at the point $x\oplus H\in G/H$, and $G/H$ is
a homogeneous $T_{1}$-space.

A topological space $X$ is called a coset space if $X$ is homeomorphic to
$G/H$, for some closed subgroup $H$ of a topological group $G$. It is well-
known that every first-countable topological group is metrizable by the
Birkhoff-Kakutani theorem. However, the following example shows that it does
not hold in coset spaces.

######  Example 3.5.

[25] The two arrows space is a compact coset space which is first-countable,
but not submetrizable.

Since every topological group is a strongly topological gyrogroup and each
subgroup is a strong subgyrogroup, the Example 3.5 shows that the axioms of
first-countability is not equivalent with metrizability in the quotient space
$G/H$, where $G$ is a strongly topological gyrogroup and $H$ is a closed
strong subgyrogroup of $G$. However, if $G$ is a strongly topological
gyrogroup with neighborhood base $\mathscr{U}$ of $0$ and $P$ is an admissible
subgyrogroup generated from $\mathscr{U}$, if the quotient space $G/P$ is
first-countable, then it is metrizable.

Indeed, in [9], Bao and Lin proved that if $G$ is a strongly topological
gyrogroup with the symmetric neighborhood base $\mathscr{U}$ at $0$ and $P$ is
an admissible $L$-subgyrogroup of $G$ generated from $\mathscr{U}$, then the
left coset space $G/P$ is submetrizable, which means that the topology
generated by a metric on $G/P$ is coarser than the quotient topology.
Therefore, it suffices to show that if the the quotient space $G/P$ is first-
countable, then the quotient topology is coarser than the topology generated
by the same metric on $G/P$. However, for convenience of reading, we write the
main process of proof about this result below.

######  Theorem 3.6.

Let $G$ be a strongly topological gyrogroup with a symmetric neighborhood base
$\mathscr{U}$ at $0$ and $P$ an admissible subgyrogroup generated from
$\mathscr{U}$. If the quotient space $G/P$ is first-countable, then it is
metrizable.

###### Proof.

Let $\\{U_{n}:n\in\mathbb{N}\\}$ be a sequence of symmetric open neighborhoods
of the identity $0$ in $G$ satisfying $U_{n}\in\mathscr{U}$ and
$U_{n+1}\oplus(U_{n+1}\oplus U_{n+1})\subseteq U_{n}$, for each
$n\in\mathbb{N}$, and such that $P=\bigcap_{n\in\mathbb{N}}U_{n}$. By [8,
Lemma 3.12], there exists a continuous prenorm $N$ on $G$ which satisfies

$N(\mbox{gyr}[x,y](z))=N(z)$

for any $x,y,z\in G$ and

$\\{x\in G:N(x)<1/2^{n}\\}\subseteq U_{n}\subseteq\\{x\in G:N(x)\leq
2/2^{n}\\},$

for any $n\in\mathbb{N}$.

It is easy to show that $N(x)=0$ if and only if $x\in P$ and $N(x\oplus
p)=N(x)$ for every $x\in G$ and $p\in P$. Then define a function $d$ from
$G\times G$ to $\mathbb{R}$ by $d(x,y)=|N(x)-N(y)|$ for all $x,y\in G$. It is
obvious that $d$ is continuous and $d$ is a pseudometric. Therefore, if
$x^{\prime}\in x\oplus P$ and $y^{\prime}\in y\oplus P$, then there exist
$p_{1},p_{2}\in P$ such that $x^{\prime}=x\oplus p_{1}$ and
$y^{\prime}=y\oplus p_{2}$, then

$d(x^{\prime},y^{\prime})=|N(x\oplus h_{1})-N(y\oplus
h_{2})|=|N(x)-N(y)|=d(x,y).$

This enables us to define a function $\varrho$ on $G/P\times G/P$ by

$\varrho(\pi_{P}(x),\pi_{P}(y))=d(\ominus x\oplus y,0)+d(\ominus y\oplus x,0)$

for any $x,y\in G$.

It is obvious that $\varrho$ is continuous, and $\varrho$ is a metric on
$G/H$.

Finally, it suffices to verify that $\varrho$ generates the quotient topology
of the space $G/H$. Given any points $x\in G$, $y\in G/H$ and any
$\varepsilon>0$, we define open balls,

$B(x,\varepsilon)=\\{x^{\prime}\in G:d(x^{\prime},x)<\varepsilon\\}$

and

$B^{*}(y,\varepsilon)=\\{y^{\prime}\in
G/H:\varrho(y^{\prime},y)<\varepsilon\\}$

in $G$ and $G/H$, respectively. Obviously, if $x\in G$ and $y=\pi_{P}(x)$,
then we have $B(x,\varepsilon)=\pi^{-1}_{P}(B^{*}(y,\varepsilon))$. Therefore,
the topology generated by $\varrho$ on $G/P$ is coarser than the quotient
topology.

Moreover, it follows from Lemma 3.4 that the quotient space $G/P$ is
homogenous. Since $G/P$ is first-countable, there exists a countable base
$\mathcal{V}=\\{V_{n}:n\in\mathbb{N}\\}$ at $\pi(0)$ in $G/P$. As $\pi$ is an
open and continuous mapping from $G$ onto $G/P$ by Lemma 3.3, $\pi(W)$ is an
open neighborhood of $\pi(0)$ for arbitrary open neighborhood $W$ of
$\pi^{-1}(\pi(0))$. We can find $V_{m}\in\mathcal{V}$ such that
$\pi(0)\subseteq V_{m}\subseteq\pi(W)$. Then
$\pi^{-1}(\pi(0))\subseteq\pi^{-1}(V_{m})\subseteq W$. Then $P$ has a
countable character in $G$. Suppose that the preimage $O=\pi^{-1}_{P}(Q)$ is
open in $G$, where $Q$ is a non-empty subset of $G/P$. For every $y\in Q$, we
have $\pi^{-1}_{P}(y)=x\oplus P\subseteq O$, where $x$ is an arbitrary point
of the fiber $\pi^{-1}_{P}(y)$. Since $\\{\pi^{-1}(V_{n}):n\in\omega\\}$ is a
base for $G$ at $P$, there exists $n\in\omega$ such that
$x\oplus\pi^{-1}(V_{n})\subseteq O$. Then there exists $\delta>0$ such that
$B(x,\delta)\subseteq x\oplus\pi^{-1}(V_{n})$. Therefore, we have
$\pi^{-1}_{P}(B^{*}(y,\delta))=B(x,\delta)\subseteq
x\oplus\pi^{-1}(V_{n})\subseteq O$. It follows that $B^{*}(y,\delta)\subseteq
Q$. So the set $Q$ is the union of a family of open balls in $(G/P,\varrho)$.
Hence, $Q$ is open in $(G/P,\varrho)$, which proves that the metric and
quotient topologies on $G/P$ coincide. ∎

By Example 3.5 and Theorem 3.6, we know that in a strongly topological
gyrogroup $G$ with a symmetric neighborhood base $\mathscr{U}$ at $0$, even
though the admissible subgyrogroup $P$ generated from $\mathscr{U}$ is a
strong subgyrogroup and both of them can be a coset to define a partition of
$G$, respectively, but the quotient spaces $G/P$ and $G/H$ have some different
properties.

######  Definition 3.7.

[14] A subgyrogroup $H$ of a topological gyrogroup $G$ is called inner (outer)
neutral if for every open neighborhood $U$ of $0$ in $G$, there exists an open
neighborhood $V$ of $0$ such that $H\oplus V\subseteq U\oplus H$ ($V\oplus
H\subseteq H\oplus U$).

It was proved in [14] that if $G$ is a strongly topological gyrogroup, $H$ is
a closed strong subgyrogroup of $G$ and $H$ is inner neutral, then $G/H$ is
first-countable if and only if $G/H$ is Fréchet-Urysohn with an
$\omega^{\omega}$-base. Here, we continue to study some properties about the
axioms of first-countability in the quotient spaces $G/H$, where $G$ is a
strongly topological gyrogroup and $H$ is a inner neutral closed strong
subgyrogroup of $G$.

######  Theorem 3.8.

Suppose that $G$ is a strongly topological gyrogroup, $H$ is a closed strong
subgyrogroup of $G$ and $H$ is inner neutral, then the followings are
equivalent.

$(1)$ $G/H$ is a sequential $\alpha_{4}$-space;

$(2)$ $G/H$ is Fréchet-Urysohn;

$(3)$ $G/H$ is strongly Fréchet-Urysohn.

###### Proof.

It suffices to prove that $(1)\Rightarrow(2)\Rightarrow(3)$. Suppose further
that the space $G/H$ is non-discrete.

$(1)\Rightarrow(2)$. For $A\subseteq G/H$, we write $[A]$ the set of all limit
points of sequences in $A$. Suppose on the contrary that $G/H$ is not Fréchet-
Urysohn. There is a subset $B$ of $G/H$ such that $[B]\neq\overline{B}$. If
$[B]$ is closed in $G/H$, then
$\overline{B}\subseteq\overline{[B]}=[B]\subseteq\overline{B}$, which is a
contradiction. Hence, $[B]$ is not closed in $G$. By the hypothesis, $G/H$ is
sequential, so $[B]$ is not sequentially closed, that is $[[B]]\neq[B]$. Thus
there is $b\in[[B]]\setminus[B]$. We may assume $b=\pi(0)$ without loss of
generality, since $G/H$ is homogeneous by Lemma 3.4.

Let $\\{b_{n}:n\in\mathbb{N}\\}$ be a sequence of points of $[B]$ converging
to $\pi(0)$. For each $n\in\mathbb{N}$, fix a point
$x_{n}\in\pi^{-1}(b_{{n}})$. For each $b_{{n}}$, let
$\\{b_{{n}}(j):j\in\mathbb{N}\\}$ be a sequence of points of $B$ converging to
$b_{{n}}$. For each $j\in\mathbb{N}$, fix a point
$x_{n}(j)\in\pi^{-1}(b_{{n}}(j))$. We claim $\lim_{j\to\infty}\pi((\ominus
x_{n})\oplus x_{n}(j))=\pi(0)$.

Indeed, let $O$ be an open neighborhood of $\pi(0)$ in $G/H$, then there is an
open neighborhood $U$ of $0$ in $G$ such that $\pi(U)\subseteq O$. Since
$\lim_{j\to\infty}\pi(x_{n}(j))=\lim_{j\to\infty}b_{n}(j)=b_{n}=\pi(x_{n})$,
there is $m\in\mathbb{N}$ such that $\pi(x_{n}(j))\in\pi(x_{n}\oplus U)$ for
$j\geq m$. So $\pi((\ominus x_{n})\oplus x_{n}(j))\in\pi((\ominus
x_{n})\oplus(x_{n}\oplus U))=\pi(U)\subseteq O$ for $j\geq m$. Hence
$\lim_{j\to\infty}\pi((\ominus x_{n})\oplus x_{n}(j))=\pi(0)$.

Since $G/H$ is an $\alpha_{4}$-space, it is possible to pick $n_{k},j_{k}$ for
each $k\in\mathbb{N}$ such that $\\{\pi((\ominus x_{n_{k}})\oplus
x_{n_{k}}(j_{k})):k\in\mathbb{N}\\}$ converges to $\pi(0)$ and $n_{k}<n_{k+1}$
for each $k\in\mathbb{N}$. It follows from [14, Lemma 3.17] that
$\lim_{k\to\infty}b_{n_{k}}(j_{k})=\lim_{k\to\infty}\pi(x_{n_{k}}(j_{k}))=\lim_{k\to\infty}\pi(x_{n_{k}}\oplus((\ominus
x_{n_{k}})\oplus x_{n_{k}}(j_{k})))=\pi(0)$, this contradicts the assumption
that $\pi(0)\not\in[B]$. Therefore, $G/H$ is Fréchet-Urysohn.

$(2)\Rightarrow(3)$. It is enough to verify condition (SFU) for $\pi(0)$.
Suppose that $\pi(0)\in\bigcap_{n\in\mathbb{N}}\overline{B_{n}}$, where each
$B_{n}$ is a subset of $G/H$. Fix a sequence $\\{x_{n}:n\in\mathbb{N}\\}$ in
$G/H\setminus\\{\pi(0)\\}$ converging to $\pi(0)$. For each $n\in\mathbb{N}$,
fix a point $a_{n}\in\pi^{-1}(x_{n})$. Since $G$ is a strong topological
gyrogroup and $H$ is inner neutral, we can fix symmetric open neighbourhoods
$U_{n}$, $V_{n}$ of $0$ such that $\pi(a_{n})\not\in\pi(U_{n}\oplus U_{n})$,
$V_{n}\subseteq U_{n}$ and $H\oplus V_{n}\subseteq U_{n}\oplus H$ for each
$n\in\mathbb{N}$. Let $A_{n}=\pi^{-1}(B_{n})$. Since $\pi$ is open continuous
by Lemma 3.3,
$0\in\bigcap_{n\in\mathbb{N}}\pi^{-1}(\overline{B_{n}})=\bigcap_{n\in\mathbb{N}}\overline{\pi^{-1}({B_{n}})}=\bigcap_{n\in\mathbb{N}}\overline{A_{n}}$.
Since $0\in\overline{A_{n}}$, we may assume that $A_{n}\subseteq V_{n}$, for
each $n\in\mathbb{N}$ (otherwise, replace $A_{n}$ with the intersection
$A_{n}\cap V_{n}$). Put $C_{n}=a_{n}\oplus A_{n}$, for $n\in\mathbb{N}$. From
the choice of $V_{n}$ it is clear that $\pi(0)\not\in\overline{\pi(C_{n})}$,
while $x_{{n}}\in\overline{\pi(C_{n})}$, for $n\in\mathbb{N}$. The last
condition and the fact that $\\{x_{{n}}:n\in\mathbb{N}\\}$ converges to
$\pi(0)$ implies that $\pi(0)\in\overline{\pi(C)}$, where
$C=\bigcup\\{C_{n}:n\in\mathbb{N}\\}$.

Since the space $G/H$ is Fréchet-Urysohn, there exists a sequence
$\eta=\\{d_{n}:n\in\mathbb{N}\\}$ in $\pi(C)$ converging to $\pi(0)$. Since
$\pi(0)$ is not in the closure of $\pi(C_{n})$, the sequence $\eta$ must
intersect $\pi(C_{n})$ for infinitely many of $n$. For every $n\in\mathbb{N}$,
choose $k_{n}\in\mathbb{N}$ such that $d_{n}\in\pi(C_{k_{n}})$. Hence, there
exists $c_{n}\in C_{k_{n}}$ such that $\pi(c_{n})=d_{n}$ for each
$n\in\mathbb{N}$. Put $b_{n}=(\ominus(a_{k_{n}}))\oplus c_{n}$, then
$\pi(b_{n})\in B_{k_{n}}$. It follows from [14, Lemma 3.17] that
$\lim_{n\to\infty}\pi(b_{n})=\pi(0)$. Thus, condition (SFU) is satisfied, and
the space $G/H$ is strongly Fréchet-Urysohn. ∎

######  Lemma 3.9.

Let $G$ be a strongly topological gyrogroup and $H$ a closed strong
subgyrogroup of $G$. If $U$ and $V$ are open neighborhoods of $0$ in $G$ with
$(\ominus V)\oplus V\subseteq U$. Then $\overline{\pi(V)}\subseteq\pi(U)$.

###### Proof.

Take any $x\in G$ such that $\pi(x)\in\overline{\pi(V)}$. Since $V\oplus x$ is
an open neighborhood of $x$, $\pi(V\oplus x)$ is an open neighborhood of
$\pi(x)$ by Lemma 3.3. Then $\pi(V\oplus x)\cap\pi(V)\not=\emptyset$. We can
find $a\in V$ and $b\in V$ such that $\pi(a\oplus x)=\pi(b)$, that is,
$a\oplus x=b\oplus h$, for some $h\in H$. Then

$\displaystyle x$ $\displaystyle=$ $\displaystyle(\ominus a)\oplus(b\oplus h)$
$\displaystyle=$ $\displaystyle((\ominus a)\oplus b)\oplus\mbox{gyr}[\ominus
a,b](h)$ $\displaystyle\in$ $\displaystyle((\ominus a)\oplus b)\oplus H$
$\displaystyle\subseteq$ $\displaystyle((\ominus V)\oplus V)\oplus H$
$\displaystyle\subseteq$ $\displaystyle U\oplus H.$

Thus, $\pi(x)\in\pi(U\oplus H)=\pi(U)$, which means that
$\overline{\pi(V)}\subseteq\pi(U)$. ∎

######  Theorem 3.10.

Let $G$ be a strongly topological gyrogroup and $H$ a closed strong
subgyrogroup of $G$. Then the quotient space $G/H$ is regular.

###### Proof.

Let $W$ be an arbitrary open neighborhood of $\pi(0)$ in $G/H$. By Lemma 3.3,
the natural quotient mapping $\pi$ is open and continuous, so there exists an
open neighborhood $U$ of $0$ in $G$ such that $\pi(U)\subseteq W$. It follows
from [4, Proposition 8] that we can find an open neighborhood $V$ of $0$ such
that $(\ominus V)\oplus V\subseteq U$. By Lemma 3.9,
$\overline{\pi(V)}\subseteq\pi(U)\subseteq W$. It is clear that $\pi(V)$ is an
open neighborhood of $\pi(0)$ in $G/H$, so we obtain that $G/H$ is regular at
the point $\pi(0)$. Moreover, it is verified that $G/H$ is homogeneous by
Lemma 3.4, hence the quotient space $G/H$ is regular. ∎

######  Theorem 3.11.

Suppose that $G$ is a strongly topological gyrogroup with neighborhood base
$\mathscr{U}$ of $0$, $H$ is a closed strong subgyrogroup of $G$ and $H$ is
inner neutral, if the space $G/H$ is bisequential, then $G/H$ is first-
countable.

###### Proof.

Since the space $G/H$ is regular by Theorem 3.10 and is also bisequential, we
can find a countable open prefilter $\xi$ on $G/H$ converging to $\pi(0)$ by
[2, Lemma 4.7.11]. Let $Q_{P}=\pi^{-1}(P)$ for each $P\in\xi$. Put
$\gamma=\\{\pi((\ominus Q_{P})\oplus Q_{P}):P\in\xi\\}$. Then $\gamma$ is a
base at $\pi(0)$ in $G/H$. Indeed, all elements of $\gamma$ are open in $G/H$
and contain $\pi(0)$. Let $O$ be an open neighborhood of $\pi(0)$. Take open
symmetric neighborhoods $U,V\in\mathscr{U}$ in $G$ such that $V\oplus
V\subseteq U$, $H\oplus(V\oplus V)\subseteq U\oplus H$ and $\pi(U)\subseteq
O$. Since $\xi$ converges to $\pi(0)$, there exists $P\in\xi$ such that
$P\subseteq\pi(V)$. It follows that $Q_{P}\subseteq V\oplus H$. Then

$\displaystyle\ominus Q_{P}$ $\displaystyle\subseteq$
$\displaystyle\ominus(V\oplus H)$ $\displaystyle=$ $\displaystyle\bigcup_{v\in
V,h\in H}\\{\ominus(v\oplus h)\\}$ $\displaystyle=$
$\displaystyle\bigcup_{v\in V,h\in H}\\{gyr[v,h](\ominus h\ominus v)\\}$
$\displaystyle=$ $\displaystyle\bigcup_{v\in V,h\in H}\\{gyr[v,h](\ominus
h)\oplus gyr[v,h](\ominus v)\\}$ $\displaystyle=$ $\displaystyle H\oplus V.$

Therefore, $0\in(\ominus Q_{P})\oplus Q_{P}\subseteq(H\oplus V)\oplus(V\oplus
H)$ and

$\displaystyle(H\oplus V)\oplus(V\oplus H)$ $\displaystyle=$
$\displaystyle\bigcup_{v_{1},v_{2}\in V,h_{1},h_{2}\in H}\\{(h_{1}\oplus
v_{1})\oplus(v_{2}\oplus h_{2})\\}$ $\displaystyle=$
$\displaystyle\bigcup_{v_{1},v_{2}\in V,h_{1},h_{2}\in H}\\{((h_{1}\oplus
v_{1})\oplus v_{2})\oplus gyr[h_{1}\oplus v_{1},v_{2}](h_{2})\\}$
$\displaystyle=$ $\displaystyle\bigcup_{v_{1},v_{2}\in V,h_{1},h_{2}\in
H}\\{(h_{1}\oplus(v_{1}\oplus gyr[v_{1},h_{1}](v_{2})))\oplus gyr[h_{1}\oplus
v_{1},v_{2}](h_{2})\\}$ $\displaystyle=$ $\displaystyle(H\oplus(V\oplus
V))\oplus H$ $\displaystyle\subseteq$ $\displaystyle(U\oplus H)\oplus H,$

hence, $\pi(0)\in\pi((\ominus Q_{P})\oplus Q_{P})\subseteq\pi((U\oplus
H)\oplus H)=\pi(U)\subseteq O$. Hence, $G/H$ is first-countable.

∎

It follows from [46, Corollary 1.3.10(2)] that every weakly first-countable
Fréchet-Urysohn space is first-countable, so it is clear that if $G$ is a
strongly topological gyrogroup, $H$ is a inner neutral and closed strong
subgyrogroup of $G$ and the space $G/H$ is weakly first-countable, then $G/H$
is first-countable. Next, we give the other type of methods to that the
equivalence between the properties of weakly first-countable and first-
countable in $G/H$.

######  Definition 3.12.

Let $\mathscr{P}$ be a family of subsets of a space $X$ with
$x\in\bigcap\mathscr{P}$.

$(1)$ The family $\mathscr{P}$ is called a network at $x$ [23] if for each
neighborhood $U$ of $x$ there exists $P\in\mathscr{P}$ such that $P\subseteq
U$.

$(2)$ The family $\mathscr{P}$ is called a $cs$-network at $x$ [42] if for any
sequence $L$ converging to $x$ and a neighborhood $U$ of $x$, there exists
$P\in\mathscr{P}$ such that $L$ is eventually in $P$ and $P\subseteq U$.

$(3)$ The family $\mathscr{P}$ is called an $sn$-network at $x$ [41] if
$\mathscr{P}$ is a network at $x$ and each element of $\mathscr{P}$ is a
sequential neighborhood of $x$.

$(4)$ The family $\mathscr{P}$ is called an $so$-network at $x$ [41] if
$\mathscr{P}$ is a network at $x$ and each element of $\mathscr{P}$ is a
sequential open subset of $X$.

$(5)$ A space $X$ is called $csf$-countable (resp., $snf$-countable,
$sof$-countable) [41] if for each $x\in X$, there is a countable $cs$-network
(resp., $sn$-network, $so$-network) at $x$.

Note that in [7], $csf$-countable spaces and $snf$-countable spaces are called
spaces with countable $cs^{\ast}$-character and spaces with countable
$sb$-character, respectively. A family $\mathscr{P}$ of subsets of a space $X$
is called a $cs$-network if for each $x\in X$, $\\{P\in\mathscr{P}:x\in P\\}$
is a $cs$-network at $x$.

According to the process of the proof of [2, Lemma 4.7.1] and [2, Proposition
4.7.2], we can easily obtain the following two lemmas:

######  Lemma 3.13.

Let $\\{V_{n}(x):n\in\omega,x\in X\\}$ and $\\{W_{n}(x):n\in\omega,x\in X\\}$
be two $sn$-networks on a Hausdorff space $X$. Then, for each $x\in X$ and
each $n\in\omega$, there is $m\in\omega$ such that $W_{m}(x)\subseteq
V_{n}(x)$.

######  Lemma 3.14.

Let $\\{V_{n}(x):n\in\omega,x\in X\\}$ be an $sn$-network on a homogeneous
Hausdorff space $X$, and let $b$ be an element of $X$. Suppose further that,
for each $x\in X$, $f_{x}$ is a homeomorphism of $X$ onto $X$ such that
$f_{x}(b)=x$. Put $W_{n}(x)=f_{x}(V_{n}(b))$. Then
$\\{W_{n}(x):n\in\omega,x\in X\\}$ is an $sn$-network on $X$.

######  Theorem 3.15.

Suppose that $G$ is a strongly topological gyrogroup with neighborhood base
$\mathscr{U}$ of $0$, $H$ is a closed strong subgyrogroup of $G$ and $H$ is
inner neutral, if the space $G/H$ is an $snf$-countable space, then $G/H$ is
an $sof$-countable space.

###### Proof.

For each $x\in G/H$, fix a point $a_{x}\in G$ such that $\pi(a_{x})=x$, where
$a_{\pi(0)}=0$. For each subset $K$ of $G/H$, put $T_{K}=\pi^{-1}(K)$.

Let $\\{O_{n}(x):n\in\omega,x\in G/H\\}$ be an $sn$-network on $G/H$. Put
$Q_{n}(x)=\pi(a_{x}\oplus(T_{O_{n}(\pi(0))}\oplus T_{O_{n}(\pi(0))}))$ for
each $x\in G/H$ and $n\in\omega$.

Claim 1: $\\{Q_{n}(x):n\in\omega,x\in G/H\\}$ is a network on $G/H$.

We can assume that $O_{n}(x)=\pi(a_{x}\oplus T_{O_{n}(\pi(0))})$ for each
$x\in G/H$ and $n\in\omega$. Indeed, this follows from Lemma 3.14 since
$h_{a}$ on $G/H$ given by the formula $h_{a}(b\oplus H)=(a\oplus b)\oplus H$,
for each $a\in G$, is a homeomorphism of $G/H$ onto itself. Next, we only need
prove that $\\{Q_{n}(x):n\in\omega,x\in G/H\\}$ is a network of $G/H$. Indeed,
let $V$ be a neighborhood of a point $x\in G/H$. Since $G$ is a strongly
topological gyrogroup and $H$ is inner neutral, there exists $W\in\mathscr{U}$
such that $\pi(a_{x}\oplus((W\oplus H)\oplus(W\oplus H)))\subseteq V$. It
follows from $\pi(0)\in\pi(W\oplus H)$ that there exists $n\in\omega$ such
that $O_{n}(\pi(0))\subseteq\pi(W\oplus H)$. So $x\in
Q_{n}(x)=\pi(a_{x}\oplus(T_{O_{n}(\pi(0))}\oplus T_{O_{n}(\pi(0))}))\subseteq
V$, whence $\\{Q_{n}(x):n\in\omega,x\in G/H\\}$ is a network of $G/H$.

Claim 2: $O_{n}(\pi(0))$ contains a sequentially open neighbourhood of
$\pi(0)$ for each $n\in\omega$.

Let $B_{n}$ be the set of all points $x\in O_{n}(\pi(0))$ such that
$\pi(a_{x}\oplus T_{O_{k}(\pi(0))})\subseteq O_{n}(\pi(0))$, for some
$k\in\omega$. Clearly, $\pi(0)\in B_{n}\subseteq O_{n}(\pi(0))$. We claim that
the set $B_{n}$ is sequentially open in $G/H$. Indeed, take any $y\in B_{n}$.
Then $\pi(a_{y}\oplus T_{O_{k}(\pi(0))})\subseteq O_{n}(\pi(0))$, for some
$k\in\omega$. By Claim 1 and Lemma 3.13, there is $m\in\omega$ such that
$\pi(T_{O_{m}(\pi(0))}\oplus T_{O_{m}(\pi(0))})\subseteq O_{k}(\pi(0))$. Then
$\pi(a_{y}\oplus(T_{O_{m}(\pi(0))}\oplus
T_{O_{m}(\pi(0))}))\subseteq\pi(a_{y}\oplus T_{O_{k}(\pi(0))})\subseteq
O_{n}(\pi(0))$, which implies that $O_{m}(y)=\pi(a_{y}\oplus
T_{O_{m}(\pi(0))})\subseteq B_{n}$. Since $y$ was an arbitrary point of
$B_{n}$, it follows that $B_{n}$ is sequentially open in $G/H$.

Now it is clear that $\\{B_{n}:n\in\omega\\}$ is a countable $so$-network of
$G/H$ at $\pi(0)$. Hence $G/H$ is an $sof$-countable space. ∎

######  Corollary 3.16.

Suppose that $G$ is a strongly topological gyrogroup, $H$ is a closed strong
subgyrogroup of $G$ and $H$ is inner neutral, if the space $G/H$ is weakly
first-countable, then $G/H$ is a first-countable space.

A partially ordered set $(T,\leq)$ is called a tree if, for each $t\in T$, the
set $\mathord{\downarrow}t=\\{s\in T:s\leq t\\}$ is well-ordered by the order
$\leq$. Given an element $t\in T$, let $\mathord{\uparrow}t=\\{s\in T:s\geq
t\\}$ and succ$(t)=\mbox{min}(\mathord{\uparrow}t\setminus\\{t\\})$ be the set
of successors of $t$ in $T$. A maximal linearly ordered subset of a tree $T$
is called a branch of $T$. By max$(T)$ (min$(T)$) we denote the set of maximal
(minimal) elements of a tree $T$. Under a sequence tree in a space $X$ we
understand a tree $(T,\leq)$ such that (1) $T\subseteq X$; (2) $T$ has no
infinite branch; (3) for each $t\not\in\mbox{max}(T)$, the set succ$(t)$ is
countable and converges to $t$.

######  Theorem 3.17.

Let $G$ be a strongly topological gyrogroup with neighborhood base
$\mathscr{U}$ of $0$, $H$ a inner neutral and closed strong subgyrogroup of
$G$. If the space $G/H$ is a $csf$-countable and sequential
$\alpha_{7}$-space, then $G/H$ is first-countable.

###### Proof.

Let $\mathcal{A}$ be a countable $cs$-network at $\pi(0)$ in $G/H$. Without
loss of generality, we may assume that the family $\mathcal{A}$ is closed
under finite unions and finite intersections. For each $x\in G/H$, fix
$a_{x}\in G$ such that $\pi(a_{x})=x$, where $a_{\pi(0)}=0$. For each subset
$K$ of $G/H$, put $T_{K}=\pi^{-1}(K)$. Since $G$ is a strongly topological
gyrogroup and $H$ is inner neutral in $G$, we can assume that
$\pi(T_{A_{1}}\oplus(T_{A_{2}}\oplus(\cdots\oplus(T_{A_{n-1}}\oplus
T_{A_{n}}))\cdots)\in\mathcal{A}$ if $A_{i}\in\mathcal{A}$ for each $i\leq n$.
It suffices to prove that the subcollection
$\mathcal{A}^{\prime}=\\{A\in\mathcal{A}:A\mbox{~{}is a sequential
neighborhood of~{}}\pi(0)\\}$ of $\mathcal{A}$ is a network at $\pi(0)$ in
$G/H$, which implies that $G/H$ has a countable $sn$-network at $\pi(0)$.

Assuming the contrary, then there exists $U\in\mathscr{U}$ such that
$A\not\subseteq p(U)$ for any $A\in\mathcal{A}^{\prime}$. Let
$\mathcal{A}^{\prime\prime}=\\{A\in\mathcal{A}:A\subseteq p(U)\\}$. Then
$\mathcal{A}^{\prime}\cap\mathcal{A}^{\prime\prime}=\emptyset$. Since
$\mathcal{A}^{\prime\prime}$ is a countable family, we may write
$\mathcal{A}^{\prime\prime}=\\{A_{n}:n\in\omega\\}$. Put $B_{n}=\bigcup_{k\leq
n}A_{k}$ for every $n\in\omega$. Note that
$B_{n}\in\mathcal{A}^{\prime\prime}$ for every $n\in\omega$.

Let $m_{-1}=0$ and $U_{-1}=U$. Since $B_{0}$ is not a sequential neighborhood
at $\pi(0)$ in $G/H$, there exists a sequence
$L_{0}=\\{x_{0,i}\\}_{i\in\omega}$ in $G/H$ such that $L_{0}$ converging to
$\pi(0)$ and $L_{0}\cap B_{0}=\emptyset$. Since the space $G/H$ is regular by
Theorem 3.10, we can choose $U_{0}\in\mathscr{U}$ such that $(U_{0}\oplus
H)\oplus(U_{0}\oplus H)\subseteq U\oplus H$ and $\overline{\pi((U_{0}\oplus
H)\oplus(U_{0}\oplus H))}\subseteq\pi(U)$. It follows that there exists
$m_{0}\in\omega$ such that $L_{0}$ is eventually in
$A_{m_{0}}\subseteq\pi(U_{0})$. Without loss of generality, we can assume that
$L_{0}\subseteq A_{m_{0}}$. By induction, we can construct $L_{k}$, $m_{k}$
and $U_{k}\in\mathscr{U}$ such that for each $k\in\omega$,

(i) $m_{k}>m_{k-1}$, $L_{k}=\\{x_{k,i}\\}_{i\in\omega}$ converges to $\pi(0)$
and $L_{k}\subseteq p(U_{k})\setminus B_{m_{k-1}}$;

(ii) $L_{k}\subseteq A_{m_{k}}\subseteq B_{m_{k}}$;

(iii) $\overline{\pi(U_{k})}\cap\\{x_{j,i}:j,i<k\\}=\emptyset$ and
$\overline{\pi((U_{k}\oplus H)\oplus(U_{k}\oplus H))}\subseteq\pi(U_{k-1})$.

Let $X=\bigcup_{k\in\omega}L_{k}$, $Y=\overline{X}\setminus X$. Then $X$ is a
discrete subspace of $G/H$. It follows that $Y$ is closed in $G/H$. Consider
the following two cases.

Case 1. $\pi(0)$ is an isolated point of $Y$.

We can find $W\in\mathscr{U}$ such that $\overline{p(W)}\cap Y=\\{\pi(0)\\}$.
It follows that $X^{\prime}=(X\cup\\{\pi(0)\\})\cap\overline{p(W)}=(X\cup
Y)\cap\overline{p(W)}=\overline{X}\cap\overline{p(W)}$ is closed in $G/H$. For
every $k\in\omega$ consider the sequence $S_{k}=L_{k}\cap\overline{p(W)}$
convergent to $\pi(0)$. Since $G/H$ is an $\alpha_{7}$-space, there is a
convergent sequence $S\subseteq G/H$ such that $S\cap S_{k}\neq\emptyset$ for
infinitely many sequences $S_{k}$. Taking into account that $X^{\prime}$ is a
closed subspace of $G/H$ with $|X^{\prime}\cap S|=\aleph_{0}$, we conclude
that the limit point lim$S$ of $S$ belongs to the set $X^{\prime}$. Moreover,
we can assume that $S\subseteq X^{\prime}$. Since the space $X$ is discrete,
lim$S\in X^{\prime}\setminus X=\\{\pi(0)\\}$. Thus the sequence $S$ converges
to $\pi(0)$. Since $\mathcal{A}^{\prime\prime}$ is a $cs$-network at $\pi(0)$
in $G/H$, there is a number $k_{0}\in\omega$ such that $S$ is eventually in
$B_{m_{k_{0}}}$. Then $S$ cannot meet infinitely many sequences $S_{k}$, this
contradicts to the choice of $S$.

Case 2. $\pi(0)$ is a cluster point of $Y$.

Since $Y$, as a closed subspace of $G/H$, is sequential, we may pick a non-
trival sequence in $Y$ converging to $\pi(0)$. Then it follows from [7, Lemma
1] that there exists a sequential tree $T\subseteq\overline{X}$ such that
min$(T)=\\{\pi(0)\\}$, max$(T)\subset X$ and succ$(\pi(0))\subseteq Y$. Let
$t_{0}=\pi(0)$. Since $\mathcal{A}$ is a $cs$-network at $\pi(0)$ in $G/H$,
there is $C_{0}\in\mathcal{A}$ such that $C_{0}\subseteq\pi(U_{0})$ and
succ$(\pi(0))$ is eventually in $C_{0}$. Pick $t_{1}\in\mbox{succ}(t_{0})\cap
C_{0}$. By induction, we can construct a finite branch $\\{t_{i}:i\leq n+1\\}$
of the tree $T$ and a sequence $\\{C_{i}:i\leq n\\}$ of elements of
$\mathcal{A}$ such that succ$(t_{i})$ is eventually in $\pi(a_{t_{i}}\oplus
T_{C_{i}})$, $C_{i}\subseteq\pi(U_{i})$ and
$t_{i+1}\in\mbox{succ}(t_{i})\cap\pi(a_{t_{i}}\oplus T_{C_{i}})$ for each
$i\leq n$. Note that the infinite set
$M=\mbox{succ}(t_{n})\cap\pi(a_{t_{n}}\oplus T_{C_{n}})$ converges to the
point $t_{n}\neq\pi(0)$. In addition, $M\subseteq\pi(a_{t_{n}}\oplus
T_{C_{n}})\subseteq\pi(a_{t_{n-1}}\oplus(T_{C_{n-1}}\oplus
T_{C_{n}}))\subseteq\cdots\subseteq\pi(a_{t_{1}}\oplus(T_{C_{1}}\oplus(\cdots\oplus(T_{C_{n-1}}\oplus
T_{C_{n}})\cdots)\subseteq\pi(T_{C_{0}}\oplus(T_{C_{1}}\oplus(\cdots\oplus(T_{C_{n-1}}\oplus
T_{C_{n}})\cdots)\subseteq\pi((U_{0}\oplus H)\oplus((U_{1}\oplus
H)\oplus(\cdots\oplus((U_{n-1}\oplus H)\oplus(U_{n}\oplus
H))\cdots)\subseteq\pi(U)$. By our assumption on $\mathcal{A}$,
$\pi(T_{C_{0}}\oplus(T_{C_{1}}\oplus(\cdots\oplus(T_{C_{n-1}}\oplus
T_{C_{n}})\cdots)\in\mathcal{A}^{\prime\prime}$ and
$\pi(T_{C_{0}}\oplus(T_{C_{1}}\oplus(\cdots\oplus(T_{C_{n-1}}\oplus
T_{C_{n}})\cdots)\subseteq B_{m_{k}}$ for some $k$. Consequently, $M\subseteq
X\cap B_{m_{k}}$ and $M\subseteq\\{x_{j,i}:j\leq k,i\in\omega\\}$ by the item
(i) of the construction of $X$. Since $\pi(0)$ is a unique cluster point of
the set $\\{x_{j,i}:j\leq k,i\in\omega\\}$, the sequence $M$ cannot converge
to $t_{n}\neq\pi(0)$, which is a contradiction.

Therefore, $G/H$ is $snf$-countable. Moreover, it is well-known that a
topological space $X$ is weakly first-countable if and only if $X$ is
sequential and $snf$-countable. Then we obtain that $G/H$ is weakly first-
countable, hence a first-countable space. We complete the proof. ∎

Therefore, we conclude the following results.

######  Corollary 3.18.

Suppose that $G$ is a strongly topological gyrogroup, $H$ is a closed strong
subgyrogroup of $G$ and $H$ is inner neutral, then the followings are
equivalent.

$(1)$ $G/H$ is first-countable;

$(2)$ $G/H$ is a bisequential space;

$(3)$ $G/H$ is a weakly first-countable space;

$(4)$ $G/H$ is a $csf$-countable and sequential $\alpha_{7}$-space.

##  4 Quotient spaces with locally compact strong subgyrogroups

In this section, it is proved that if $G$ is a strongly topological gyrogroup
and $H$ is a locally compact strong subgyrogroup of $G$, then there exists an
open neighborhood $U$ of the identity element $0$ such that
$\pi(\overline{U})$ is closed in $G/H$ and the restriction of $\pi$ to
$\overline{U}$ is a perfect mapping from $\overline{U}$ onto the subspace
$\pi(\overline{U})$, which implies that if $H$ is a locally compact strong
subgyrogroup of a strongly topological gyrogroup $G$ and the quotient space
$G/H$ has some nice properties, such as locally compact, locally countably
compact, locally pseudocompact, etc., then $G$ also has the same properties.

######  Proposition 4.1.

If $H$ is a locally compact subgyrogroup of a topological gyrogroup $G$, then
$H$ is closed in $G$.

###### Proof.

We show that $H=\overline{H}$ in $G$. By [4, Proposition 7], $\overline{H}$ is
a subgyrogroup of $G$. Since $H$ is a dense locally compact subspace of
$\overline{H}$, we know that $H$ is open in $\overline{H}$. Therefore, $H$ is
closed, which means that $H=\overline{H}$. Therefore, $H$ is a closed
subgyrogroup in $G$. ∎

######  Lemma 4.2.

Let $G$ be a topological gyrogroup. For each open neighborhood $U$ of $0$ and
$x\in G$, there exists an open neighborhood $V$ of $0$ such that $V\oplus
x\subseteq x\oplus U$.

###### Proof.

For each open neighborhood $U$ of $0$ and $x\in G$, define $L_{\ominus
x}(y):G\rightarrow G$ by $L_{\ominus x}(y)=\ominus x\oplus y$ and
$R_{x}:G\rightarrow G$ by $R_{x}(y)=y\oplus x$. For each open neighborhood $U$
of $0$, we can find an open neighborhood $W$ of $x$ such that $L_{\ominus
x}(W)=(\ominus x)\oplus W\subseteq U$, as $L_{\ominus x}$ is continuous. By
the same method, we can find an open neighborhood $V$ of $0$ such that
$R_{x}(V)=V\oplus x\subseteq W$. Then $(\ominus x)\oplus(V\oplus x)\subseteq
U$, which means that $V\oplus x\subseteq x\oplus U$. ∎

######  Theorem 4.3.

Let $G$ be a strongly topological gyrogroup with neighborhood base
$\mathscr{U}$ of $0$, $F$ a compact subset of $G$, and $P$ a closed subset of
$G$ such that $F\cap P=\emptyset$. Then there exists an open neighborhood $V$
of the identity element $0$ such that $(F\oplus V)\cap P=\emptyset$ and
$(V\oplus F)\cap P=\emptyset$.

###### Proof.

For each $x\in F$, we can find an open neighborhood $W_{x}$ of $0$ in $G$ such
that $(x\oplus W_{x})\cap P=\emptyset$. Then choose an open neighborhood
$O_{x}\in\mathscr{U}$ with $O_{x}\oplus O_{x}\subseteq W_{x}$. Since $F$ is
compact and $F\subseteq\bigcup_{x\in F}\\{x\oplus O_{x}\\}$, we can find a
finite subset $C\subseteq F$ such that $F\subseteq\bigcup_{x\in C}\\{x\oplus
O_{x}\\}$. Put $Q=\bigcap_{x\in C}O_{x}$. For each $y\in F$, there exists
$x\in C$ with $y\in x\oplus O_{x}$. Then $y\oplus Q\subseteq(x\oplus
O_{x})\oplus Q\subseteq(x\oplus O_{x})\oplus O_{x}=\bigcup_{s,t\in
O_{x}}\\{(x\oplus s)\oplus t\\}=\bigcup_{s,t\in
O_{x}}\\{x\oplus(s\oplus\mbox{gyr}[s,x](t))\\}=x\oplus(O_{x}\oplus
O_{x})\subseteq x\oplus W_{x}\subseteq G\setminus P$. Therefore, $(F\oplus
Q)\cap P=\emptyset$.

On the other hand, choose $Q_{1}\in\mathscr{U}$ such that $Q_{1}\oplus
Q_{1}\subseteq Q$. By Lemma 4.2, for each $y\in F$, we can find an open
neighborhood $U_{y}$ of $0$ such that $U_{y}\oplus y\subseteq y\oplus Q_{1}$.
Then $F\subseteq\bigcup_{y\in F}\\{U_{y}\oplus y\\}$. Since $F$ is compact,
there is a finite set $E\subseteq F$ such that $F\subseteq\bigcup_{y\in
E}\\{U_{y}\oplus y\\}$. Put $U=\bigcap_{y\in E}U_{y}$. For each $t\in F$, we
can find $y\in E$ with $t\in U_{y}\oplus y$. Then

$\displaystyle U\oplus t$ $\displaystyle\subseteq$ $\displaystyle
U\oplus(U_{y}\oplus y)$ $\displaystyle\subseteq$ $\displaystyle
U\oplus(y\oplus Q_{1})$ $\displaystyle=$
$\displaystyle\bigcup\\{u\oplus(y\oplus q):u\in U,q\in Q_{1}\\}$
$\displaystyle=$ $\displaystyle\bigcup\\{(u\oplus
y)\oplus\mbox{gyr}[u,y](q):u\in U,q\in Q_{1}\\}$ $\displaystyle=$
$\displaystyle(U\oplus y)\oplus Q_{1}$ $\displaystyle\subseteq$
$\displaystyle(U_{y}\oplus y)\oplus Q_{1}$ $\displaystyle\subseteq$
$\displaystyle(y\oplus Q_{1})\oplus Q_{1}$ $\displaystyle=$
$\displaystyle\bigcup\\{(y\oplus p)\oplus q:p,q\in Q_{1}\\}$ $\displaystyle=$
$\displaystyle\bigcup\\{y\oplus(p\oplus\mbox{gyr}[p,y](q)):p,q\in Q_{1}\\}$
$\displaystyle=$ $\displaystyle y\oplus(Q_{1}\oplus Q_{1})$
$\displaystyle\subseteq$ $\displaystyle y\oplus Q$ $\displaystyle\subseteq$
$\displaystyle G\setminus P.$

Therefore, $(U\oplus F)\cap P=\emptyset$. Finally, put $V=Q\cap U$, and we
obtain that $(F\oplus V)\cap P=\emptyset$ and $(V\oplus F)\cap P=\emptyset$. ∎

######  Lemma 4.4.

Let $G$ be a strongly topological gyrogroup and $H$ a locally compact strong
subgyrogroup of $G$. Suppose that $P$ is a closed symmetric subset of $G$ such
that $P$ contains an open neighborhood of $0$ in $G$, and
$\overline{P\oplus(P\oplus P)}\cap H$ is compact. Then the restriction $f$ of
$\pi$ to $P$ is a perfect mapping from $P$ onto the subspace $\pi(P)$ of
$G/H$.

###### Proof.

It is obvious that $f$ is continuous. Moreover, since $H$ is a locally compact
subgyrogroup of $G$, we know that $H$ is closed in $G$ by Proposition 4.1.

Claim 1. $f^{-1}(f(a))$ is compact for each $a\in P$.

Indeed, from the definition of $f$, we have $f^{-1}(f(a))=(a\oplus H)\cap P$.
By [4, Proposition 3], the left gyrotranslation is a homeomorphism, so the
subspace $(a\oplus H)\cap P$ and $H\cap((\ominus a)\oplus P)$ are
homeomorphic, thus both of them are closed in $G$. From $\ominus a\in\ominus
P=P$, it follows that

$H\cap((\ominus a)\oplus P)\subseteq H\cap(P\oplus
P)\subseteq\overline{P\oplus(P\oplus P)}\cap H.$

Hence, $H\cap((\ominus a)\oplus P)$ is compact and so is the set $f^{-1}f(a)$.

Claim 2. $f$ is a closed mapping.

Let us fix any closed subset $M$ of $P$ and let $a$ be an any point of $P$
such that $f(a)\in\overline{f(M)}$. It suffices to show that $f(a)\in f(M)$.

Suppose on the contrary. Then $(a\oplus H)\cap(M\oplus H)\cap P=\emptyset$.
Since $H$ is a strong subgyrogroup in $G$, $(a\oplus H)\oplus
H=\bigcup_{x,y\in H}\\{(a\oplus x)\oplus y\\}=\bigcup_{x,y\in
H}\\{a\oplus(x\oplus\mbox{gyr}[x,a](y))\\}=a\oplus(H\oplus H)=a\oplus H$. Then
$(a\oplus H)\cap M\cap P=\emptyset$, and $(a\oplus H)\cap\overline{(P\oplus
P)}\cap M=\emptyset$ since $M\subseteq P$. Obviously, $(a\oplus
H)\cap\overline{(P\oplus P)}$ is compact. Since $M$ is a closed and disjoint
from the compact subset $(a\oplus H)\cap\overline{(P\oplus P)}$, by Theorem
4.3, there exists an open neighborhood $W\in\mathscr{U}$ such that $W\subseteq
P$ and $(W\oplus((a\oplus H)\cap\overline{(P\oplus P)}))\cap M=\emptyset$.

Since the quotient mapping $\pi$ is open by Lemma 3.3 and $W\oplus a$ is an
open neighborhood of $a$, the set $\pi(W\oplus a)$ is an open neighborhood of
$\pi(a)$ in $G/H$. Therefore, the set $\pi(W\oplus a)\cap\pi(M)\neq\emptyset$
and we can fix $m\in M$ and $y\in W$ such that $\pi(m)=\pi(y\oplus a)$, that
is, $m\in(y\oplus a)\oplus H$. Then, $(y\oplus a)\oplus H=y\oplus(a\oplus
gyr[a,y](H))=y\oplus(a\oplus H)$. Hence, there exists an $h\in H$ such that
$a\oplus h=\ominus y\oplus m$. Since $\ominus y\in\ominus W=W\subseteq P$ and
$m\in M\subseteq P$, we have that $a\oplus h=\ominus y\oplus
m\in\overline{(P\oplus P)}$. In addition, $a\oplus h\in a\oplus H$. Hence,
$a\oplus h\in((a\oplus H)\cap\overline{(P\oplus P)})$ and $m\in
W\oplus((a\oplus H)\cap\overline{(P\oplus P)})$. Thus, $M\cap(W\oplus((a\oplus
H)\cap\overline{(P\oplus P)}))\neq\emptyset$, which is a contradiction.

Therefore, $f(a)\in f(M)$ and $f(M)$ is closed in $f(P)$. Then, since $a\in P$
is arbitrarily taken, we conclude that the mapping $f$ is perfect. ∎

The following result is very important in this paper.

######  Theorem 4.5.

Let $G$ be a strongly topological gyrogroup and $H$ a locally compact strong
subgyrogroup of $G$. Then there exists an open neighborhood $U$ of the
identity element $0$ such that $\pi(\overline{U})$ is closed in $G/H$ and the
restriction of $\pi$ to $\overline{U}$ is a perfect mapping from
$\overline{U}$ onto the subspace $\pi(\overline{U})$.

###### Proof.

Since $H$ is locally compact, we know that $H$ is closed in $G$ by Proposition
4.1 and we can find an open neighborhood $V$ of $0$ in $G$ such that
$\overline{V\cap H}$ is compact. Since $G$ is regular, we can choose an open
neighborhood $W$ of $0$ such that $\overline{W}\subseteq V$. As a closed
subspace of the compact set $\overline{V\cap H}$, $\overline{W}\cap H$ is
compact. Let $U_{0}$ be an arbitrary symmetric open neighborhood of $0$ such
that $U_{0}\oplus(U_{0}\oplus U_{0})\subseteq W$. By the joint continuity, we
have
$\overline{U_{0}}\oplus(\overline{U_{0}}\oplus\overline{U_{0}})\subseteq\overline{U_{0}\oplus(U_{0}\oplus
U_{0})}$. Then the set $P=\overline{U_{0}}$ satisfies all restrictions on $P$
in Lemma 4.4. It follows from Lemma 4.4 that the restriction of $\pi$ to $P$
is a perfect mapping from $P$ onto the subspace $\pi(P)$.

It follows from Lemma 3.3 that $\pi$ is an open mapping, the set $\pi(U_{0})$
is open in $G/H$. It follows from Theorem 3.10 that the space $G/H$ is
regular, then we can find an open neighborhood $V_{0}$ of $\pi(0)$ in $G/H$
such that $\overline{V_{0}}\subseteq\pi(U_{0})$. Hence $U=\pi^{-1}(V_{0})\cap
U_{0}$ is an open neighborhood of $0$ contained in $P$ such that the
restriction $f$ of $\pi$ to $\overline{U}$ is a perfect mapping from
$\overline{U}$ onto the subspace $\pi(\overline{U})$. Furthermore,
$\pi(\overline{U})$ is closed in $\pi(P)$, and
$\pi(\overline{U})\subseteq\overline{V_{0}}\subseteq\pi(U_{0})\subseteq\pi(P)$.
Then $\pi(\overline{U})$ is closed in $\overline{V_{0}}$, so that
$\pi(\overline{U})$ is closed in $G/H$. ∎

######  Corollary 4.6.

Let $\mathscr{P}$ be a topological property preserved by preimages of spaces
under perfect mappings (in the class of completely regular spaces) and also
inherited by regular closed sets. Let $G$ be a strongly topological gyrogroup
and $H$ a locally compact strong subgyrogroup of $G$. Then if the quotient
space $G/H$ has the property $\mathscr{P}$, we can find an open neighborhood
$U$ of the identity element $0$ such that $\overline{U}$ has the property
$\mathscr{P}$.

As we all know, local compactness, countable compactness, pseudocompactness,
the Lindelöf property, $\sigma$-compactness and $\check{\rm C}$ech-
completeness are all inherited by regular closed sets and preserved by perfect
preimages. Then the followings are clear by Corollary 4.6.

######  Corollary 4.7.

Let $G$ be a strongly topological gyrogroup and $H$ a locally compact strong
subgyrogroup of $G$. If the quotient space $G/H$ has some of the following
properties:

(1) $G/H$ is locally compact;

(2) $G/H$ is locally countably compact;

(3) $G/H$ is locally pseudocompact;

(4) $G/H$ is locally $\sigma$-compact;

(5) $G/H$ is locally Lindelöf;

(6) $G/H$ is locally $\check{C}$ech-complete; and

(7) $G/H$ is locally realcompact,
then $G$ also has the same property.

Since paracompactness is inherited by regular closed sets and preserved by
perfect preimages and it was proved in [10, Theorem 4.6] that every locally
paracompact strongly topological gyrogroup is paracompact, the following
result is trivial.

######  Corollary 4.8.

Let $G$ be a strongly topological gyrogroup and $H$ a locally compact strong
subgyrogroup of $G$. If the quotient space $G/H$ is locally paracompact, then
$G$ is a paracompact space.

######  Corollary 4.9.

Let $G$ be a strongly topological gyrogroup and $H$ a locally compact strong
subgyrogroup of $G$. If the quotient space $G/H$ is a $k$-space, then $G$ is
also a $k$-space.

######  Proof.

Since the property of being a $k$-space is invariant under taking perfect
preimages and a locally $k$-space is a $k$-space, see [23, Section 3.3], it
follows that $G$ is also a $k$-space. ∎

A topological gyrogroup is feathered if it contains a non-empty compact set
$K$ of countable character in $G$.

######  Lemma 4.10.

[12] Let $G$ be a strongly topological gyrogroup. Then the followings are
equivalent:

(1) $G$ is feathered,

(2) $G$ is a $p$-space, and

(3) $G$ is a paracompact $p$-space.

######  Theorem 4.11.

Let $G$ be a strongly topological gyrogroup and $H$ a locally compact strong
subgyrogroup of $G$. If the quotient space $G/H$ is a feathered space, then
$G$ is a paracompact $p$-space.

###### Proof.

By Theorem 4.5, there exists an open neighborhood $U$ of the identity element
$0$ in $G$ such that $\overline{U}$ is a preimage of a closed subset of $G/H$
under a perfect mapping. Moreover, since the class of feathered spaces is
closed under taking closed subspaces, it follows from [2, Proposition 4.3.36]
that $\overline{U}$ is a feathered space. Therefore, $U$ contains a non-empty
compact subspace $F$ with a countable base of neighborhoods in $G$, thus $G$
is a paracompact $p$-space by Lemma 4.10. ∎

##  5 Quotient spaces with locally compact and metrizable strong subgyrogroups

In this section, we give some applications about Theorem 4.5 combining
generalized metric properties. In particular, we assume that the strong
subgyrogroup $H$ of a strongly topological gyrogroup $G$ is locally compact
and metrizable.

######  Definition 5.1.

[52] Let $X$ be a topological space. A space is called strictly Fréchet-
Urysohn at a point $x\in X$ if whenever $\\{A_{n}\\}_{n}$ is a sequence of
subsets in $X$ and $x\in\bigcap_{n\in\mathbb{N}}\overline{A_{n}}$, there
exists $x_{n}\in A_{n}$ for each $n\in\mathbb{N}$ such that the sequence
$\\{x_{n}\\}_{n}$ converges to $x$. A space $X$ is called strictly Fréchet-
Urysohn if it is strictly Fréchet-Urysohn at every point $x\in X$.

######  Lemma 5.2.

[44] Suppose that $X$ is a regular space, and that $f:X\rightarrow Y$ is a
closed mapping. Suppose also that $b\in X$ is a $G_{\delta}$-point in the
space $F=f^{-1}(f(b))$ (i.e., the singleton $\\{b\\}$ is a $G_{\delta}$-set in
the space $F$) and $F$ is countably compact and strictly Fréchet-Urysohn at
$b$. If the space $Y$ is strictly Fréchet-Urysohn at $f(b)$, then $X$ is
strictly Fréchet-Urysohn at $b$.

######  Theorem 5.3.

Let $G$ be a strongly topological gyrogroup and $H$ a locally compact
metrizable strong subgyrogroup of $G$. If the quotient space $G/H$ is strictly
Fréchet-Urysohn, then $G$ is also strictly Fréchet-Urysohn.

###### Proof.

By Theorem 4.5, there exists an open neighborhood $U$ of the identity element
$0$ in $G$ such that
$\pi|_{\overline{U}}:\overline{U}\rightarrow\pi(\overline{U})$ is a perfect
mapping and $\pi(\overline{U})$ is closed in $G/H$.

Put $f=\pi|_{\overline{U}}:\overline{U}\rightarrow\pi(\overline{U})$. Then
$f(\overline{U})=\pi(\overline{U})$ is strictly Fréchet-Urysohn. For each
$b\in\overline{U}$, $f^{-1}(f(b))=\pi^{-1}(\pi(b))\cap\overline{U}=(b\oplus
H)\cap\overline{U}$ is compact and metrizable. It follows from Lemma 5.2 that
$\overline{U}$ is strictly Fréchet-Urysohn. Therefore, $G$ is locally strictly
Fréchet-Urysohn and $G$ is strictly Fréchet-Urysohn. ∎

######  Lemma 5.4.

[2, Proposition 4.7.18] Suppose that $X$ is a regular space, and that
$f:X\rightarrow Y$ is a closed mapping. Suppose also that $b\in X$ is a
$G_{\delta}$-point in the space $F=f^{-1}(f(b))$ (i.e., the singleton
$\\{b\\}$ is a $G_{\delta}$-set in the space $F$) and $F$ is Fréchet-Urysohn
at $b$. If the space $Y$ is strongly Fréchet-Urysohn, then $X$ is Fréchet-
Urysohn at $b$.

######  Theorem 5.5.

Let $G$ be a strongly topological gyrogroup and $H$ a locally compact
metrizable strong subgyrogroup of $G$. If the quotient space $G/H$ is strongly
Fréchet-Urysohn, then the space $G$ is also strongly Fréchet-Urysohn.

###### Proof.

By Theorem 4.5, there exists an open neighborhood $U$ of the identity element
$0$ in $G$ such that
$\pi|_{\overline{U}}:\overline{U}\rightarrow\pi(\overline{U})$ is a perfect
mapping and $\pi(\overline{U})$ is closed in $G/H$.

Put $f=\pi|_{\overline{U}}:\overline{U}\rightarrow\pi(\overline{U})$. Then
$f(\overline{U})=\pi(\overline{U})$ is strongly Fréchet-Urysohn. For each
$b\in\overline{U}$, $f^{-1}(f(b))=\pi^{-1}(\pi(b))\cap\overline{U}=(b\oplus
H)\cap\overline{U}$ is metrizable. Therefore, the singleton $\\{b\\}$ is a
$G_{\delta}$-set in the space $f^{-1}(f(b))$. Moreover, since the quotient
space $G/H$ is strongly Fréchet-Urysohn, the space $G$ is locally Fréchet-
Urysohn by Lemma 5.4. Hence, $G$ is Fréchet-Urysohn. Furthermore, every
Fréchet-Urysohn topological gyrogroup is strongly Fréchet-Urysohn by [37,
Corollary 5.2]. So $G$ is strongly Fréchet-Urysohn. ∎

######  Theorem 5.6.

Let $G$ be a strongly topological gyrogroup and $H$ a locally compact
metrizable strong subgyrogroup of $G$. If the quotient space $G/H$ is
sequential, then $G$ is also sequential.

###### Proof.

By Theorem 4.5, there exists an open neighborhood $U$ of the identity element
$0$ in $G$ such that
$\pi|_{\overline{U}}:\overline{U}\rightarrow\pi(\overline{U})$ is a perfect
mapping and $\pi(\overline{U})$ is closed in $G/H$.

First, we show that if $\\{x_{n}\\}_{n}$ is a sequence in $\overline{U}$ such
that $\\{\pi(x_{n})\\}_{n}$ is a convergent sequence in $\pi(\overline{U})$
and $x$ is an accumulation point of the sequence $\\{x_{n}\\}_{n}$, then there
is a subsequence of $\\{x_{n}\\}_{n}$ which converges to $x$.

Since $\pi|_{\overline{U}}$ is perfect, every subsequence of $\\{x_{n}\\}_{n}$
has an accumulation point in $\overline{U}$. Put
$F=\pi^{-1}(\pi(x))\cap\overline{U}$. Since $H$ is metrizable,
$\pi^{-1}(\pi(x))=x\oplus H$ is also metrizable. Since every topological
gyrogroup is regular, there exists a sequence $\\{U_{k}\\}_{k}$ of open
subsets in $G$ such that $\overline{U_{k+1}}\subseteq U_{k}$ for each
$k\in\mathbb{N}$ and $\\{x\\}=F\cap\bigcap_{k\in\mathbb{N}}U_{k}$. Choose a
subsequence $\\{x_{n_{k}}\\}_{k}$ of $\\{x_{n}\\}_{n}$ such that $x_{n_{k}}\in
U_{k}$ for each $k\in\mathbb{N}$. For an arbitrary accumulation point $p$ of a
subsequence of the sequence $\\{x_{n_{k}}\\}_{k}$, we have $\pi(p)=\pi(x)$ and
$p\in\bigcap_{k\in\mathbb{N}}\overline{U_{k}}$. Thus $p=x$. Therefore, $x$ is
the unique accumulation point of every subsequence of $\\{x_{n_{k}}\\}_{k}$,
proving that $x_{n_{k}}\rightarrow x$.

Then choose an open neighborhood $V$ of $0$ such that $\overline{V}\subseteq
U$ and we show that $\overline{V}$ is a sequential subspace.

Suppose that $\overline{V}$ is not a sequential subspace, so we can find a
non-closed and sequentially closed subset $A$ of $\overline{V}$. Then there
exists a point $x$ such that $x\in cl_{\overline{V}}(A)\setminus A$. It is
clear that $cl_{\overline{V}}(A)=\overline{A}$. Let
$f=\pi|_{\overline{V}}:\overline{V}\rightarrow\pi(\overline{V})$ and $B=A\cap
f^{-1}(f(x))$. Since $B$ is a closed subset of $A$, $B$ is sequentially
closed. Moreover, the fiber $f^{-1}(f(x))=(\pi^{-1}(\pi(x)))\cap\overline{V}$
is sequential, so $B$ is closed in $\overline{V}$. Since $x\not\in B$, there
exists an open neighborhood $W$ of $x$ in $\overline{V}$ such that
$\overline{W}\cap B=\emptyset$. Let $C=\overline{W}\cap A$, then $C$ is also
sequentially closed as a closed subset of $A$ and $x\in\overline{C}\setminus
C$. Therefore, $C\cap f^{-1}(f(x))=\overline{W}\cap B=\emptyset$, then
$f(x)\in\overline{f(C)}\setminus f(C)$. So $f(C)=\pi(C)$ is not closed in
$\pi(\overline{V})$. However, this is impossible, as it is easy to verify that
the image of each sequentially closed subset of $\overline{V}$ is closed in
$\pi(\overline{V})$.

Indeed, let $C$ be sequentially closed in $\overline{V}$ and $\\{y_{n}\\}_{n}$
a sequence in $\pi(C)$ such that $y_{n}\rightarrow y$ in $\pi(\overline{V})$.
Choose $x_{n}\in C$ with $\pi(x_{n})=y_{n}$ for each $n\in\mathbb{N}$. Since
every subsequence of the sequence $\\{x_{n}\\}_{n}$ has an accumulation point,
there exist a point $x\in\pi^{-1}(y)$ and a subsequence $\\{x_{n_{k}}\\}_{k}$
of $\\{x_{n}\\}_{n}$ such that $x_{n_{k}}\rightarrow x$. Since $C$ is
sequentially closed, we obtain $x\in C$ and $y\in\pi(C)$. Therefore, $\pi(C)$
is sequentially closed in $\pi(\overline{V})$. Since
$\pi|_{\overline{U}}:\overline{U}\rightarrow\pi(\overline{U})$ is a closed
mapping and $\pi(\overline{U})$ is closed in $G/H$, $\pi(\overline{V})$ is
closed in $G/H$. Since $G/H$ is sequential, $\pi(\overline{V})$ is also
sequential and then $\pi(C)$ is closed in $\pi(\overline{V})$.

Since $G$ is homogeneous and $\overline{V}$ is a sequential subspace, we
conclude that $G$ is a locally sequential space. Thus, $G$ is a sequential
space. ∎

However, for the property of Fréchet-Urysohn, we do not know whether it has
the similar result. Therefore, we pose the following question.

######  Question 5.7.

Let $G$ be a strongly topological gyrogroup and $H$ a locally compact
metrizable strong subgyrogroup of $G$. If the quotient space $G/H$ is Fréchet-
Urysohn, is $G$ also Fréchet-Urysohn?

######  Theorem 5.8.

Let $G$ be a strongly topological gyrogroup and $H$ a locally compact
metrizable strong subgyrogroup of $G$. If the quotient space $G/H$ has
property $\mathcal{P}$, where $\mathcal{P}$ is a topological property. Then
the space $G$ is locally in $\mathcal{P}$ if $\mathcal{P}$ satisfies the
following:

(1) $\mathcal{P}$ is closed hereditary;

(2) $\mathcal{P}$ contains point $G_{\delta}$-property, and

(3) let $f:X\rightarrow Y$ be a perfect mapping, if $X$ has
$G_{\delta}$-diagonal and $Y$ is $\mathcal{P}$, then $X$ is $\mathcal{P}$.

###### Proof.

By the hypothesis, since $G/H$ is in $\mathcal{P}$ and $\mathcal{P}$ contains
point $G_{\delta}$-property, $\\{H\\}$ is a $G_{\delta}$-subset in $G/H$, that
is, there exists a sequence $\\{V_{n}:n\in\mathbb{N}\\}$ of open sets in $G/H$
such that $\\{H\\}=\bigcap_{n\in\mathbb{N}}V_{n}$. Therefore,
$H=\bigcap_{n\in\mathbb{N}}\pi^{-1}(V_{n})$. Since $H$ is a metrizable strong
subgyrogroup of $G$, there is a family $\\{W_{n}:n\in\mathbb{N}\\}$ of open
neighborhoods of the identity element $0$ such that $\\{W_{n}\cap
H:n\in\mathbb{N}\\}$ is an open countable neighborhood base in $H$. Hence,

$\\{0\\}=\bigcap_{n\in\mathbb{N}}(W_{n}\cap
H)=\bigcap_{n\in\mathbb{N}}(W_{n}\cap\pi^{-1}(V_{n})).$

Then $G$ has point $G_{\delta}$-property. It follows from [9] that every
strongly topological gyrogroup with countable pseudocharacter is
submetrizable. So $G$ has $G_{\delta}$-diagonal.

By Theorem 4.5, there is an open neighborhood $U$ of the identity element $0$
in $G$ such that
$\pi|_{\overline{U}}:\overline{U}\rightarrow\pi(\overline{U})$ is a perfect
mapping and $\pi(\overline{U})$ is closed in $G/H$. Then by (1) and (3), the
subspace $\overline{U}$ is in $\mathcal{P}$. Therefore, $G$ is locally in
$\mathcal{P}$. ∎

It is well-known that all stratifiable spaces, semi-stratifiable spaces and
$\sigma$-spaces satisfy the conditions in Theorem 5.8, respectively. Moreover,
it was claimed in [13] and [43] that if a strongly topological gyrogroup $G$
has point $G_{\delta}$-property, then $G$ has a $KG$-sequence and if
$f:X\rightarrow Y$ is a perfect map and $Y$ is a $k$-semistratifiable space,
then $X$ is a $k$-semistratifiable space if and only if $X$ has a
$KG$-sequence. Therefore, the following corollary is obtained.

######  Corollary 5.9.

Let $G$ be a strongly topological gyrogroup and $H$ a locally compact
metrizable strong subgyrogroup of $G$. If the quotient space $G/H$ is a
stratifiable space (semi-stratifiable space, $k$-semistratifiable,
$\sigma$-space), then $G$ is a local stratifiable space (semi-stratifiable
space, $k$-semistratifiable, $\sigma$-space).

Finally, we pose the following questions.

######  Question 5.10.

Let $G$ be a strongly topological gyrogroup and $H$ a closed strong
subgyrogroup of $G$. Is the quotient space $G/H$ completely regular?

######  Question 5.11.

Let $\mathcal{P}$ be any class of topological spaces which is closed
hereditary and closed under locally finite unions of closed sets. Is every
strongly topological gyrogroup which is locally in $\mathcal{P}$ in
$\mathcal{P}$ ?

##  6 Quotient spaces with closed first-countable and separable strong
subgyrogroups

In this section, we study the quotient space $G/H$ with some generalized
metric properties, where $G$ is a strongly topological gyrogroup and $H$ is a
closed first-countable and separable strong subgyrogroup of $G$. In
particular, we prove that if the quotient space $G/H$ is an
$\aleph_{0}$-space, then $G$ is an $\aleph_{0}$-space; if the quotient space
$G/H$ is a cosmic space, then $G$ is also a cosmic space; if the quotient
space $G/H$ has a star-countable $cs$-network or star-countable
$wcs^{*}$-network, then $G$ also has a star-countable $cs$-network or star-
countable $wcs^{*}$-network, respectively.

######  Definition 6.1.

[33, 40] Let $\mathcal{P}$ be a family of subsets of a topological space $X$.

1\. $\mathcal{P}$ is called a k-network for $X$ if whenever $K\subseteq U$
with $K$ compact and $U$ open in $X$, there exists a finite family
$\mathcal{P}^{{}^{\prime}}\subseteq\mathcal{P}$ such that
$K\subseteq\bigcup\mathcal{P}^{{}^{\prime}}\subseteq U$.

2\. $\mathcal{P}$ 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 $\\{x_{n_{i}}:i\in\mathbb{N}\\}\subseteq
P\subseteq U$ for some $P\in\mathcal{P}$.

######  Definition 6.2.

[49] 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 $\aleph_{0}$-space if it is a regular space with a
countable $k$-network.

It was claimed in [45] that every base is a $k$-network and a $cs$-network for
a topological space, 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 and only if $X$ has a
countable $k$-network if and only if $X$ has a countable $wcs^{*}$-network,
see [44]. Therefore, it is natural that a topological space is an
$\aleph_{0}$-space if and only if it is a regular space with a countable
$cs$-network. Moreover, every $\aleph_{0}$-space is a cosmic space and every
cosmic space is a paracompact, separable space.

The following lemmas are necessary.

######  Lemma 6.3.

[13] Suppose that $G$ is a topological gyrogroup and $H$ is a closed and
separable $L$-subgyrogroup of $G$. If $Y$ is a separable subset of $G/H$,
$\pi^{-1}(Y)$ is also separable in $G$.

######  Lemma 6.4.

[10] Every locally paracompact strongly topological gyrogroup is paracompact.

######  Lemma 6.5.

[18] Every star-countable family $\mathcal{P}$ of subsets of a topological
space $X$ can be expressed as
$\mathcal{P}=\bigcup\\{\mathcal{P}_{\alpha}:\alpha\in\Lambda\\}$, where each
subfamily $\mathcal{P}_{\alpha}$ is countable and
$(\bigcup\mathcal{P}_{\alpha})\cap(\bigcup\mathcal{P}_{\beta})=\emptyset$
whenever $\alpha\not=\beta$.

######  Theorem 6.6.

Let $G$ be a strongly topological gyrogroup and $H$ a closed first-countable
and separable strong subgyrogroup of $G$. If the quotient space $G/H$ is a
local $\aleph_{0}$-space, then $G$ is a topological sum of
$\aleph_{0}$-subspace.

###### Proof.

Let $G$ be a strongly topological gyrogroup with a symmetric neighborhood base
$\mathscr{U}$ at $0$. Since the quotient space $G/H$ is a local
$\aleph_{0}$-space, we can find an open neighborhood $Y$ of $H$ in $G/H$ such
that $Y$ has a countable $cs$-network. Put $X=\pi^{-1}(Y)$. By Lemma 3.3, the
natural homomorphism $\pi$ from $G$ onto $G/H$ is an open and continuous
mapping, so $X$ is an open neighborhood of the identity element $0$ in $G$.
Since $Y$ is an $\aleph_{0}$-space and each $\aleph_{0}$-space is separable,
it follows from Lemma 6.3 that $X$ is separable. Therefore, there is countable
subset $B=\\{b_{m}:m\in\mathbb{N}\\}$ of $X$ such that $\overline{B}=X$.

Since $H$ is first-countable, there exists a countable family
$\\{V_{n}:n\in\mathbb{N}\\}\subseteq\mathscr{U}$ of open symmetric
neighborhoods of $0$ in $G$ such that $V_{n+1}\oplus(V_{n+1}\oplus
V_{n+1})\subseteq V_{n}\subseteq X$ for each $n\in\mathbb{N}$ and the family
$\\{V_{n}\cap H:n\in\mathbb{N}\\}$ is a local base at $0$ for $H$. Since $Y$
is an $\aleph_{0}$-space, there is a countable $cs$-network
$\\{P_{k}:k\in\mathbb{N}\\}$ for $Y$.

Claim 1. $X$ is an $\aleph_{0}$-space.

Put $\mathcal{F}=\\{\pi^{-1}(P_{k})\cap(b_{m}\oplus
V_{n}):k,m,n\in\mathbb{N}\\}$. Then $\mathcal{F}$ is a countable family of
subsets of $X$. Suppose that $\\{x_{i}\\}_{i}$ is a sequence converging to a
point $x$ in $X$ and $U$ be a neighborhood of $x$ in $X$. Then $U$ is also a
neighborhood of $x$ in $G$. Let $V$ be an open neighborhood of $0$ in $G$ such
that $x\oplus(V\oplus V)\subseteq U$. Since $\\{V_{n}\cap H:n\in\mathbb{N}\\}$
is a local base at $0$ for $H$, there is $n\in\mathbb{N}$ such that $V_{n}\cap
H\subseteq V\cap H$. Moreover, $(x\oplus V_{n+1})\cap X$ is a non-empty open
subset of $X$ and $\overline{B}=X$, whence $B\cap(x\oplus
V_{n+1})\not=\emptyset$. Therefore, there exists $b_{m}\in B$ such that
$b_{m}\in x\oplus V_{n+1}$. Furthermore, $(x\oplus V_{n+1})\cap(x\oplus V)$ is
an open neighborhood of $x$ and $\pi:G\rightarrow G/H$ is an open mapping, so
$\pi((x\oplus V_{n+1})\cap(x\oplus V))$ is an open neighborhood of $\pi(x)$ in
the space $Y$ and the sequence $\\{\pi(x_{i})\\}_{i}$ converges to $\pi(x)$ in
$Y$. It is obtained that

$\\{\pi(x)\\}\cup\\{\pi(x_{i}):i\geq i_{0}\\}\subseteq
P_{k}\subseteq\pi((x\oplus V_{n+1})\cap(x\oplus V))\mbox{ for some
}i_{0},k\in\mathbb{N}.$

By the left cancellation law of Lemma 2.3, it is easy to verify that $(x\oplus
V_{n+1})\cap(x\oplus V)=x\oplus(V_{n+1}\cap V)$. Therefore, for an arbitrary
$z\in\pi^{-1}(P_{k})\cap(b_{m}\oplus V_{n+1})$, $\pi(z)\in
P_{k}\subseteq\pi(x\oplus(V_{n+1}\cap V))$. Since $z\in(x\oplus(V_{n+1}\cap
V))\oplus H$, and $H$ is a strong subgyrogroup, then

$z\in(x\oplus(V_{n+1}\cap V))\oplus H=\bigcup_{t\in V_{n+1}\cap V}\\{(x\oplus
t)\oplus H\\}=\bigcup_{t\in V_{n+1}\cap
V}\\{x\oplus(t\oplus\mbox{gyr}[t,x](H))\\}=x\oplus((V_{n+1}\cap V)\oplus H).$

Therefore, $\ominus x\oplus z\in(V_{n+1}\cap V)\oplus H$. Moreover, since
$z\in b_{m}\oplus V_{n+1}$ and $b_{m}\in x\oplus V_{n+1}$, it follows that

$\displaystyle z$ $\displaystyle\in$ $\displaystyle(x\oplus V_{n+1})\oplus
V_{n+1}$ $\displaystyle=$ $\displaystyle\bigcup_{u,v\in V_{n+1}}\\{(x\oplus
u)\oplus v\\}$ $\displaystyle=$ $\displaystyle\bigcup_{u,v\in
V_{n+1}}\\{x\oplus(u\oplus\mbox{gyr}[u,x](v))\\}$ $\displaystyle=$
$\displaystyle x\oplus(V_{n+1}\oplus V_{n+1}).$

So, $(\ominus x)\oplus z\in V_{n+1}\oplus V_{n+1}$. Hence, $(\ominus x)\oplus
z\in((V_{n+1}\cap V)\oplus H)\cap(V_{n+1}\oplus V_{n+1})$. There exist
$a\in(V_{n+1}\cap V),~{}h\in H$ and $u_{3},v_{3}\in V_{n+1}$ such that
$(\ominus x)\oplus z=a\oplus h=u_{3}\oplus v_{3}$, whence $h=(\ominus
a)\oplus(u_{3}\oplus v_{3})\in V_{n+1}\oplus(V_{n+1}\oplus V_{n+1})\subseteq
V_{n}$. Therefore, $(\ominus x)\oplus z\in(V_{n+1}\cap V)\oplus(V_{n}\cap H)$,
and consequently, $z\in x\oplus((V_{n+1}\cap V)\oplus(V_{n}\cap H))\subseteq
x\oplus(V\oplus V)\subseteq U$. Thus, we obtain that
$\pi^{-1}(P_{k})\cap(b_{m}\oplus V_{n+1})\subseteq U$.

Since $b_{m}\in x\oplus V_{n+1}$, there is $u\in V_{n+1}$ such that
$b_{m}=x\oplus u$, whence

$\displaystyle x$ $\displaystyle=$ $\displaystyle(x\oplus
u)\oplus\mbox{gyr}[x,u](\ominus u)$ $\displaystyle=$ $\displaystyle
b_{m}\oplus\mbox{gyr}[x,u](\ominus u)$ $\displaystyle\in$ $\displaystyle
b_{m}\oplus\mbox{gyr}[x,u](V_{n+1})$ $\displaystyle=$ $\displaystyle
b_{m}\oplus V_{n+1}.$

Therefore, there exists $i_{1}\geq i_{0}$ such that $x_{i}\in b_{m}\oplus
V_{n+1}$ when $i\geq i_{1}$, whence $\\{x\\}\cup\\{x_{i}:i\geq
i_{1}\\}\subseteq\pi^{-1}(P_{k})\cap(b_{m}\oplus V_{n+1})$. Thus $\mathcal{F}$
is a countable $cs$-network for $X$, and hence $X$ is an $\aleph_{0}$-space.

Since $G$ is homogeneous, it is clear that $G$ is a local $\aleph_{0}$-space.
Therefore, $G$ is a locally paracompact space. Furthermore, every locally
paracompact strongly topological gyrogroup is paracompact by Lemma 6.4, so $G$
is paracompact. Let $\mathcal{A}$ be an open cover of $G$ by
$\aleph_{0}$-subspace. Because the property of being an $\aleph_{0}$-space is
hereditary, we can assume that $\mathcal{A}$ is locally finite in $G$ by the
paracompactness of $G$. Moreover, as every point-countable family of open
subsets in a separable space is countable, the family $\mathcal{A}$ is star-
countable. Then
$\mathcal{A}=\bigcup\\{\mathcal{B}_{\alpha}:\alpha\in\Lambda\\}$ by Lemma 6.5,
where each subfamily $\mathcal{B}_{\alpha}$ is countable and
$(\bigcup\mathcal{B}_{\alpha})\cap(\bigcup\mathcal{B}_{\beta})=\emptyset$
whenever $\alpha\not=\beta$. Set $X_{\alpha}=\bigcup\mathcal{B}_{\alpha}$ for
each $\alpha\in\Lambda$. Then $G=\bigoplus_{\alpha\in\Lambda}X_{\alpha}$.

Claim 2. $X_{\alpha}$ is an $\aleph_{0}$-subspace for each $\alpha\in\Lambda$.

Put $\mathcal{B}_{\alpha}=\\{B_{\alpha,n}:n\in\mathbb{N}\\}$, where each
$B_{\alpha,n}$ is an open $\aleph_{0}$-subspace of $G$, and put
$\mathcal{P}_{\alpha}=\bigcup_{n\in\mathbb{N}}\mathcal{P}_{\alpha,n}$, where
$\mathcal{P}_{\alpha,n}$ is a countable $cs$-network for the
$\aleph_{0}$-space $B_{\alpha,n}$ for each $n\in\mathbb{N}$. Then
$\mathcal{P}_{\alpha}$ is a countable $cs$-network for $X_{\alpha}$. Thus,
$X_{\alpha}$ is an $\aleph_{0}$-space.

In conclusion, $G$ is a topological sum of $\aleph_{0}$-subspaces. ∎

######  Corollary 6.7.

Let $G$ be a strongly topological gyrogroup and $H$ a closed first-countable
and separable strong subgyrogroup of $G$. If the quotient space $G/H$ is an
$\aleph_{0}$-space, $G$ is also an $\aleph_{0}$-space.

By the similar proof of Theorem 6.6, the following result is obvious.

######  Theorem 6.8.

Let $G$ be a strongly topological gyrogroup and $H$ a closed first-countable
and separable strong subgyrogroup of $G$. If the quotient space $G/H$ is a
locally cosmic space, then $G$ is a topological sum of cosmic subspaces.

######  Corollary 6.9.

Let $G$ be a strongly topological gyrogroup and $H$ a closed first-countable
and separable strong subgyrogroup of $G$. If the quotient space $G/H$ is a
cosmic space, $G$ is also a cosmic space.

######  Theorem 6.10.

Let $G$ be a strongly topological gyrogroup and $H$ a closed first-countable
and separable strong subgyrogroup of $G$. If the quotient space $G/H$ has a
star-countable $cs$-network, $G$ also has a star-countable $cs$-network.

###### Proof.

Let $\mathscr{U}$ be a symmetric neighborhood base at $0$ such that
$gyr[x,y](U)=U$ for any $x,y\in G$ and $U\in\mathscr{U}$. Since the
subgyrogroup $H$ of $G$ is first-countable at the identity element $0$ of $G$,
there exists a countable family
$\\{V_{n}:n\in\mathbb{N}\\}\subseteq\mathscr{U}$ such that
$(V_{n+1}\oplus(V_{n+1}\oplus V_{n+1}))\subseteq V_{n}$ for each
$n\in\mathbb{N}$ and the family $\\{V_{n}\cap H:n\in\mathbb{N}\\}$ is a local
base at $0$ for $H$.

Let $\mathcal{P}=\\{P_{\alpha}:\alpha\in\Lambda\\}$ be a star-countable
$cs$-network for the space $G/H$. For each $\alpha\in\Lambda$, the family
$\\{P_{\alpha}\cap P_{\beta}:\beta\in\Lambda\\}$ is a countable
$wcs^{*}$-network for $P_{\alpha}$. Therefore, $P_{\alpha}$ is a cosmic space,
and $P_{\alpha}$ is separable. Then it follows from Lemma 6.3 that
$\pi^{-1}(P_{\alpha})$ is separable. We can find a countable subset
$B_{\alpha}=\\{b_{\alpha,m}:m\in\mathbb{N}\\}$ of $\pi^{-1}(P_{\alpha})$ such
that $\overline{B_{\alpha}}=\pi^{-1}(P_{\alpha})$.

Put

$\mathcal{F}=\\{\pi^{-1}(P_{\alpha})\cap(b_{\alpha,m}\oplus
V_{n}):\alpha\in\Lambda,~{}~{}and~{}~{}m,n\in\mathbb{N}\\}.$

Then $\mathcal{F}$ is a star-countable family of $G$.

Claim. $\mathcal{F}$ 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 $(x\oplus(V\oplus V))\subseteq U$. Since $\\{V_{n}\cap
H:n\in\mathbb{N}\\}$ is a local base at $0$ for $H$, there exists
$n\in\mathbb{N}$ such that $V_{n}\cap H\subseteq V\cap H$. Since
$\pi:G\rightarrow G/H$ is an open and continuous mapping, there are
$i_{0}\in\mathbb{N}$ and $\alpha\in\Lambda$ such that
$\\{\pi(x)\\}\cup\\{\pi(x_{i}):i\geq i_{0}\\}\subseteq
P_{\alpha}\subseteq\pi((x\oplus V_{n+1})\cap(x\oplus V))$. Since
$x\in\pi^{-1}(P_{\alpha})$, $(x\oplus V_{n+1})\cap\pi^{-1}(P_{\alpha})$ is
non-empty and open in the subspace $\pi^{-1}(P_{\alpha})$. Moreover, since
$\overline{B_{\alpha}}=\pi^{-1}(P_{\alpha})$, there exists $m\in\mathbb{N}$
such that $b_{\alpha,m}\in x\oplus V_{n+1}$.

For an arbitrary $z\in\pi^{-1}(P_{\alpha})\cap(b_{\alpha,m}\oplus V_{n+1})$,
$\pi(z)\in P_{\alpha}\subseteq\pi((x\oplus V_{n+1})\cap(x\oplus
V))=\pi(x\oplus(V_{n+1}\cap V))$. Then, $z\in x\oplus((V_{n+1}\cap V)\oplus
H)$ since $H$ is a strong subgyrogroup. Since $z\in b_{\alpha,m}\oplus
V_{n+1}$ and $b_{\alpha,m}\in x\oplus V_{n+1}$, we have

$\displaystyle z$ $\displaystyle\in$ $\displaystyle(x\oplus V_{n+1})\oplus
V_{n+1}$ $\displaystyle=$ $\displaystyle\bigcup_{u,v\in V_{n+1}}\\{(x\oplus
u)\oplus v\\}$ $\displaystyle=$ $\displaystyle\bigcup_{u,v\in
V_{n+1}}\\{x\oplus(u\oplus\mbox{gyr}[u,x](v))\\}$ $\displaystyle=$
$\displaystyle x\oplus(V_{n+1}\oplus V_{n+1}).$

Then, $(\ominus x)\oplus z\in V_{n+1}\oplus V_{n+1}$. Hence, $(\ominus
x)\oplus z\in((V_{n+1}\cap V)\oplus H)\cap(V_{n+1}\oplus V_{n+1})$. Therefore,
there exist $a\in(V_{n+1}\cap V),~{}h\in H$ and $u_{1},u_{2}\in V_{n+1}$ such
that $(\ominus x)\oplus z=a\oplus h=u_{1}\oplus u_{2}$, whence $h=(\ominus
a)\oplus(u_{1}\oplus u_{2})\in V_{n+1}\oplus(V_{n+1}\oplus V_{n+1})\subseteq
V_{n}$. It follows that $(\ominus x)\oplus z\in(V_{n+1}\cap V)\oplus(V_{n}\cap
H)$. Thus $z\in x\oplus((V_{n+1}\cap V)\oplus(V_{n}\cap H))\subseteq
x\oplus(V\oplus V)\subseteq U$. Hence,
$\pi^{-1}(P_{\alpha})\cap(b_{\alpha,m}\oplus V_{n+1})\subseteq U$.

Since $b_{\alpha,m}\in x\oplus V_{n+1}$, there is $u_{3}\in V_{n+1}$ such that
$b_{\alpha,m}=x\oplus u_{3}$. Thus,

$\displaystyle x$ $\displaystyle=$ $\displaystyle(x\oplus
u_{3})\oplus\mbox{gyr}[x,u_{3}](\ominus u_{3})$ $\displaystyle=$
$\displaystyle b_{\alpha,m}\oplus\mbox{gyr}[x,u_{3}](\ominus u_{3})$
$\displaystyle\in$ $\displaystyle
b_{\alpha,m}\oplus\mbox{gyr}[x,u_{3}](V_{n+1})$ $\displaystyle=$
$\displaystyle b_{\alpha,m}\oplus V_{n+1}.$

Therefore, there exists $i_{1}\geq i_{0}$ such that $x_{i}\in
b_{\alpha,m}\oplus V_{n+1}$ whenever $i\geq i_{1}$, whence
$\\{x\\}\cup\\{x_{i}:i\geq
i_{1}\\}\subseteq\pi^{-1}(P_{\alpha})\cap(b_{\alpha,m}\oplus V_{n+1})$.

In conclusion, $G$ has a star-countable $cs$-network. ∎

######  Theorem 6.11.

Let $G$ be a strongly topological gyrogroup and $H$ a closed first-countable
and separable strong subgyrogroup of $G$. If the quotient space $G/H$ has a
star-countable $wcs^{*}$-network, $G$ also has a star-countable
$wcs^{*}$-network.

###### Proof.

Let $\mathscr{U}$ be a symmetric neighborhood base at $0$ such that
$gyr[x,y](U)=U$ for any $x,y\in G$ and $U\in\mathscr{U}$. Since the
subgyrogroup $H$ of $G$ is first-countable at the identity element $0$ of $G$,
there exists a countable family
$\\{V_{n}:n\in\mathbb{N}\\}\subseteq\mathscr{U}$ in $G$ such that
$(V_{n+1}\oplus(V_{n+1}\oplus V_{n+1}))\subseteq V_{n}$ for each
$n\in\mathbb{N}$ and the family $\\{V_{n}\cap H:n\in\mathbb{N}\\}$ is a local
base at $0$ for $H$.

We construct $\mathcal{P}$ and $\mathcal{F}$ by the same way in Theorem 6.10,
and we show that $\mathcal{F}$ 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 $(x\oplus(V\oplus V))\subseteq U$. Since $\\{V_{n}\cap
H:n\in\mathbb{N}\\}$ is a local base at $0$ for $H$, there exists
$n\in\mathbb{N}$ such that $V_{n}\cap H\subseteq V\cap H$. Since $\mathcal{P}$
is a $wcs^{*}$-network for $G/H$, there exists a subsequence
$\\{\pi(x_{i_{j}})\\}_{j}$ of the sequence $\\{\pi(x_{i})\\}_{i}$ such that
$\\{\pi(x_{i_{j}}):j\in\mathbb{N}\\}\subseteq P_{\alpha}\subseteq\pi((x\oplus
V_{n+1})\cap(x\oplus V))$ for some $\alpha\in\Lambda$. As the sequence
$\\{x_{i}\\}_{i}$ converges to $x$, we have some $x_{i_{j}}\in x\oplus
V_{n+2}$ for each $j\in\mathbb{N}$. Furthermore, since
$x_{i_{1}}\in\pi^{-1}(P_{\alpha})$, $(x_{i_{1}}\oplus
V_{n+2})\cap\pi^{-1}(P_{\alpha})$ is non-empty and open in
$\pi^{-1}(P_{\alpha})$. Then it follows from
$\overline{B_{\alpha}}=\pi^{-1}(P_{\alpha})$ that there exists
$m\in\mathbb{N}$ such that $b_{\alpha,m}\in x_{i_{1}}\oplus V_{n+2}$. Then

$\displaystyle b_{\alpha,m}$ $\displaystyle\in$ $\displaystyle x_{i_{1}}\oplus
V_{n+2}$ $\displaystyle\subseteq$ $\displaystyle(x\oplus V_{n+2})\oplus
V_{n+2}$ $\displaystyle=$ $\displaystyle\bigcup_{u,v\in V_{n+2}}\\{(x\oplus
u)\oplus v\\}$ $\displaystyle=$ $\displaystyle\bigcup_{u,v\in
V_{n+2}}\\{x\oplus(u\oplus\mbox{gyr}[u,x](v))\\}$ $\displaystyle=$
$\displaystyle x\oplus(V_{n+2}\oplus V_{n+2}).$

Moreover, it is proved in Theorem 6.10 that
$\pi^{-1}(P_{\alpha})\cap(b_{\alpha,m}\oplus V_{n+1})\subseteq U$.

In conclusion, $G$ has a star-countable $wcs^{*}$-network. ∎

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