# Embedding of Strongly Topological Gyrogroups in Path-Connected and Locally Path-Connected Gyrogroups

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1**

**.**A nonempty set G, together with a binary operation ⊕ on G, is called a gyrogroup if it satisfies the following axioms:

- (G1)
- There exists an element $e\in G$ such that $e\oplus a=a$ for all $a\in G$.
- (G2)
- For each $a\in G$, there exists an element $b\in G$ such that $b\oplus a=e$.
- (G3)
- For all a, $b\in G$, there is an automorphism $\mathrm{gyr}\left[a,b\right]\in AutG$ such that$$a\oplus (b\oplus c)=(a\oplus b)\oplus \mathrm{gyr}\left[a,b\right]c$$
- (G4)
- For all a, $b\in G$, $\mathrm{gyr}\left[a\oplus b,b\right]=\mathrm{gyr}\left[a,b\right]$. (left loop property)

**Definition**

**2**

**.**A gyrogroup G is gyrocommutative if $a\oplus b=\mathrm{gyr}[a,b](b\oplus a)$ for all $a,b\in G$.

**Definition**

**3**

**.**Let G be a gyrogroup. An element $x\in G$ is a two-torsion element if $x\oplus x=e$. If the only two-torsion element in G is e, we say that G is two-torsion-free.

**Definition**

**4**

**.**A gyrogroup G is two-divisible if for each element $x\in G$, there is an element $y\in G$ such that $y\oplus y=x$.

**Definition**

**5**

**.**Let G be a gyrogroup. An element $a\in G$ is said to have finite order if there is a number $n\in \mathbb{N}$ such that $na=e$. If $na\ne e$ for all $n\in \mathbb{N}$, a is said to have infinite order. The order of a is the least positive integer k (if any) such that $ka=e$, denoted by $\left|a\right|$. In the case when a has infinite order, we write $\left|a\right|=\infty $.

**Definition**

**6**

**.**A gyrogroup $(G,\oplus )$ endowed with a topology is called a topological gyrogroup if the following statements hold:

- (T1)
- The binary operation $\oplus :G\times G\to G$ is continuous. Here, $G\times G$ is equipped with the product topology.
- (T2)
- The operation of taking inverses, $x\mapsto \ominus x$, called the inversion function, is continuous.

## 3. Extended Gyrogroups and Their Properties

**Proof.**

**Proposition**

**1.**

**Proof.**

**Theorem**

**2.**

- (i)
- being gyrocommutative;
- (ii)
- being two-torsion-free;
- (iii)
- being two-divisible;
- (iv)
- all elements having finite order.

**Proof.**

- (i)
- Suppose that G is gyrocommutative. Let $f,g\in {G}^{\u2022}$ and let $r\in J$. Then,$$(f\oplus g)\left(r\right)=f\left(r\right)\oplus g\left(r\right)=\mathrm{gyr}\left[f\right(r),g(r\left)\right]\left(g\right(r)\oplus f(r\left)\right)=\mathrm{gyr}[f,g](g\oplus f)\left(r\right).$$Thus, $f\oplus g=\mathrm{gyr}[f,g](g\oplus f)$, and so ${G}^{\u2022}$ is gyrocommutative. The converse holds since G is isomorphic to a subgyrogroup of ${G}^{\u2022}$.
- (ii)
- Suppose that G is two-torsion-free. Let ${e}^{\u2022}\ne f\in {G}^{\u2022}$. Then, there is a number $r\in J$ such that $f\left(r\right)\ne e$. It follows that $(f\oplus f)\left(r\right)=f\left(r\right)\oplus f\left(r\right)\ne e$. Thus, $f\oplus f\ne {e}^{\u2022}$. This proves that ${G}^{\u2022}$ is two-torsion-free. Conversely, assume that ${G}^{\u2022}$ is two-torsion-free. Let $e\ne x\in G$. Then, ${x}^{\u2022}\ne {e}^{\u2022}$, and so ${x}^{\u2022}\oplus {x}^{\u2022}\ne {e}^{\u2022}$. It follows that $x\oplus x=({x}^{\u2022}\oplus {x}^{\u2022})\left(0\right)\ne e$. This proves that G is two-torsion-free.
- (iii)
- Suppose that G is two-divisible. Let $f\in {G}^{\u2022}$ and let $f\left(J\right)=\{{z}_{1},{z}_{2},\cdots ,{z}_{n}\}$. By assumption, for each $i\in \{1,2,\cdots ,n\}$, there is an element ${x}_{i}\in G$ such that ${x}_{i}\oplus {x}_{i}={z}_{i}$. Define a function $g:J\to G$ by $g\left(r\right)={x}_{i}$ for $r\in {f}^{-1}\left(\left\{{z}_{i}\right\}\right)$ for all $i=1,2,\dots ,n$. Clearly, $g\in {G}^{\u2022}$. By definition of g, $(g\oplus g)\left(r\right)=f\left(r\right)$ for all $r\in J$. Thus, ${G}^{\u2022}$ is two-divisible. Conversely, assume that ${G}^{\u2022}$ is two-divisible. Let $x\in G$. By assumption, there is a function $f\in {G}^{\u2022}$ such that $f\oplus f={x}^{\u2022}$. Thus, $f\left(0\right)\oplus f\left(0\right)=(f\oplus f)\left(0\right)={x}^{\u2022}\left(0\right)=x$. Hence, G is two-divisible.
- (iv)
- Suppose that all the elements of G have finite order. Let $f\in {G}^{\u2022}$ and let $f\left(J\right)=\{{z}_{1},{z}_{2},\cdots ,{z}_{n}\}$. Since $\left(\right|{z}_{1}\left|\right|{z}_{2}|\cdots |{z}_{n}\left|\right)f={e}^{\u2022}$, all the elements of ${G}^{\u2022}$ have finite order. The converse holds since G is isomorphic to a subgyrogroup of ${G}^{\u2022}$. ☐

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

## 4. Embedding of Strongly Topological Gyrogroups

**Theorem**

**5.**

- (i)
- for every $U\in \mathcal{U}$, there is a set $V\in \mathcal{U}$ such that $V\oplus V\subseteq U$;
- (ii)
- for all $U\in \mathcal{U},x\in G$, there is a set $V\in \mathcal{U}$ such that $x\oplus (\ominus (V\oplus x\left)\right)\subseteq U$;
- (iii)
- for all $U\in \mathcal{U},x\in U$, there is a set $V\in \mathcal{U}$ such that $x\oplus V\subseteq U$;
- (iv)
- for all $U\in \mathcal{U},x,y\in G$, there is a set $V\in \mathcal{U}$ such that $(x\oplus V)\oplus y\subseteq (x\oplus y)\oplus U$, $x\oplus (V\oplus y)\subseteq (x\oplus y)\oplus U$, and $(x\oplus V)\u229ey\subseteq (x\u229ey)\oplus U$;
- (v)
- for all $U,V\in \mathcal{U}$, there is a set $W\in \mathcal{U}$ such that $W\subseteq U\cap V$;
- (vi)
- $\bigcap \mathcal{U}=\left\{e\right\}$.

**Proof.**

- (i)
- Let $U\in \mathcal{U}$. Since the operation ⊕ is continuous and $e\oplus e\in U$, there are sets ${O}_{1},{O}_{2}\in \mathcal{U}$ such that ${O}_{1}\oplus {O}_{2}\subseteq U$. Since $\mathcal{U}$ is an open base at e, there is a set $V\in \mathcal{U}$ such that $V\subseteq {O}_{1}\cap {O}_{2}$. Hence, $V\oplus V\subseteq {O}_{1}\oplus {O}_{2}\subseteq U$.
- (ii)
- Let $U\in \mathcal{U}$ and let $x\in G$. Since G is a topological gyrogroup, the function ${L}_{x}\circ \ominus \circ {R}_{x}$ is continuous. Note that $({L}_{x}\circ \ominus \circ {R}_{x})\left(e\right)=x\oplus (\ominus (e\oplus x))=e\in U$. By the continuity of ${L}_{x}\circ \ominus \circ {R}_{x}$, there is a set $V\in \mathcal{U}$ such that $x\oplus (\ominus (V\oplus x))=({L}_{x}\circ \ominus \circ {R}_{x})\left(V\right)\subseteq U$.
- (iii)
- Let $U\in \mathcal{U}$ and let $x\in U$. Then, $\ominus x\oplus U$ is an open neighborhood of e for ${L}_{\ominus x}$ is open. Since $\mathcal{U}$ is an open base at e, there is a set $V\in \mathcal{U}$ such that $V\subseteq \ominus x\oplus U$. Thus, $x\oplus V\subseteq U$.
- (iv)
- Let $U\in \mathcal{U}$ and let $x,y\in G$. Since G is a topological gyrogroup, the function ${\left({R}_{\ominus y}\right)}^{-1}\circ {L}_{x}$ is continuous. Note that ${\left({R}_{\ominus y}\right)}^{-1}\left(z\right)=z\u229ey$ for all $z\in G$. Note also that $(x\u229ey)\oplus U$ is an open neighborhood of $x\u229ey$. Since $({\left({R}_{\ominus y}\right)}^{-1}\circ {L}_{x})\left(e\right)=(x\oplus e)\u229ey=x\u229ey\in (x\u229ey)\oplus U$, there is a set ${V}_{1}\in \mathcal{U}$ such that $(x\oplus {V}_{1})\u229ey=({\left({R}_{\ominus y}\right)}^{-1}\circ {L}_{x})\left({V}_{1}\right)\subseteq (x\u229ey)\oplus U$. Similarly, one can show that there are sets ${V}_{2},{V}_{3}\in \mathcal{U}$ such that $(x\oplus {V}_{2})\oplus y\subseteq (x\oplus y)\oplus U$ and $x\oplus ({V}_{3}\oplus y)\subseteq (x\oplus y)\oplus U$. Since $\mathcal{U}$ is an open base at e, there is a set $V\in \mathcal{U}$ such that $V\subseteq {V}_{1}\cap {V}_{2}\cap {V}_{3}$, and the assertion follows.
- (v)
- The assertion follows directly from the definition of an open base.
- (vi)
- Clearly, $\left\{e\right\}\subseteq \bigcap \mathcal{U}$. Let $e\ne x\in G$. Since G is ${T}_{1}$, there is a set $V\in \mathcal{U}$ such that $x\notin V$. Thus, $x\notin \bigcap \mathcal{U}$. This shows that $\bigcap \mathcal{U}=\left\{e\right\}$. ☐

**Definition**

**7**

**.**A topological gyrogroup G is strong if there exists an open base $\mathcal{U}$ at the identity e of G such that $\mathrm{gyr}[x,y]\left(U\right)=U$ for all $x,y\in G,U\in \mathcal{U}$. In this case, we say that G is a strongly topological gyrogroup with open base $\mathcal{U}$ at e.

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Theorem**

**6.**

- (i)
- for every $U\in \mathcal{U}$, there is a set $V\in \mathcal{U}$ such that $V\oplus V\subseteq U$;
- (ii)
- for all $U\in \mathcal{U},x\in G$, there is a set $V\in \mathcal{U}$ such that $x\oplus (\ominus (V\oplus x\left)\right)\subseteq U$;
- (iii)
- for all $U\in \mathcal{U},x\in U$, there is a set $V\in \mathcal{U}$ such that $x\oplus V\subseteq U$;
- (iv)
- for all $U\in \mathcal{U},x,y\in G$, there is a set $V\in \mathcal{U}$ such that $(x\oplus V)\oplus y\subseteq (x\oplus y)\oplus U$, $x\oplus (V\oplus y)\subseteq (x\oplus y)\oplus U$, and $(x\oplus V)\u229ey\subseteq (x\u229ey)\oplus U$;
- (v)
- for all $U,V\in \mathcal{U}$, there is a set $W\in \mathcal{U}$ such that $W\subseteq U\cap V$;
- (vi)
- $\bigcap \mathcal{U}=\left\{e\right\}$;
- (vii)
- for all $U\in \mathcal{U},x,y\in G$, $\mathrm{gyr}[x,y]\left(U\right)\subseteq U$.

**Proof.**

**Claim 1.**

**Claim 2.**

**Claim 3.**

**Claim 4.**

**Claim 5.**

**Claim 6.**

**Corollary**

**1.**

**Proof.**

**Theorem**

**7.**

**Proof.**

- (i)
- Let $O(V,\u03f5)\in \mathcal{N}\left({e}^{\u2022}\right)$. Then, there is a set $U\in \mathcal{N}\left(e\right)$ such that $U\oplus U\subseteq V$. Let $f,g\in O(U,\frac{\u03f5}{2})$. Note that $\{r\in J\mid f(r)\oplus g(r)\notin V\}\subseteq \{r\in J\mid f(r)\notin U\}\cup \{r\in J\mid g(r)\notin U\}.$ Hence,$$\begin{array}{cc}\hfill \mu \left(\right\{r\in J\mid f\left(r\right)\oplus g\left(r\right)\notin V\left\}\right)& \le \mu \left(\right\{r\in J\mid f\left(r\right)\notin U\left\}\right)+\mu \left(\right\{r\in J\mid g\left(r\right)\notin U\left\}\right)\hfill \\ \hfill & <\frac{\u03f5}{2}+\frac{\u03f5}{2}\hfill \\ \hfill & =\u03f5.\hfill \end{array}$$This shows that $f\oplus g\in O(V,\u03f5)$.
- (ii)
- Let $O(V,\u03f5)\in \mathcal{N}\left({e}^{\u2022}\right)$ and let $f\in {G}^{\u2022}$. Then, there is a partition $\{{a}_{0},{a}_{1},\cdots ,{a}_{n}\}$ of J such that f is constant on each interval $[{a}_{k},{a}_{k+1})$. By item (ii) of Theorem 5, for each $k=0,1,\cdots ,n-1$, there is a set ${U}_{k}\in \mathcal{N}\left(e\right)$ such that $f\left({a}_{k}\right)\oplus (\ominus ({U}_{k}\oplus f\left({a}_{k}\right)))\subseteq V$. By item (v) of Theorem 5, there is a set $U\in \mathcal{N}\left(e\right)$ such that $U\subseteq {\displaystyle \bigcap _{i=0}^{n-1}}{U}_{i}$. Thus, $f\left(r\right)\oplus (\ominus (U\oplus f\left(r\right)\left)\right)\subseteq V$ for all $r\in J$. Let $h\in O(U,\u03f5)$. Then,$$\begin{array}{cc}\hfill \{r\in J\mid (f\oplus (\ominus (h\oplus f\left)\right)\left)\right(r)\notin V\}& =\{r\in J\mid f(r)\oplus (\ominus \left(h\right(r)\oplus f(r\left)\right))\notin V\}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \subseteq \{r\in J\mid h(r)\notin U\}.\hfill \end{array}$$It follows that $\mu \left(\right\{r\in J\mid (f\oplus (\ominus (h\oplus f)\left)\right)\left(r\right)\notin V\left\}\right)\le \mu \left(\right\{r\in J\mid h\left(r\right)\notin U\left\}\right)<\u03f5$, and so $f\oplus (\ominus (h\oplus f\left)\right)\in O(V,\u03f5)$. This shows that $f\oplus (\ominus (O(U,\u03f5)\oplus f\left)\right)\subseteq O(V,\u03f5)$.
- (iii)
- Let $O(V,\u03f5)\in \mathcal{N}\left({e}^{\u2022}\right),f\in O(V,\u03f5)$. Then, there is a partition $\{{a}_{0},{a}_{1},\cdots ,{a}_{n}\}$ of J such that f is constant on each interval $[{a}_{k},{a}_{k+1})$. Set$$L=\{k\in \{0,1,\cdots ,n-1\}\mid f\left({a}_{k}\right)\in V\}.$$By item (iii) of Theorem 5, for each $k\in L$, there is a set ${U}_{k}\in \mathcal{N}\left(e\right)$ such that $f\left({a}_{k}\right)\oplus {U}_{k}\subseteq V$. By item (v) of Theorem 5, there is a set $U\in \mathcal{N}\left(e\right)$ such that $U\subseteq \bigcap _{k\in L}{U}_{k}$. Thus, $f\left(r\right)\oplus U\subseteq V$ whenever $f\left(r\right)\in V$. Put $\delta =\u03f5-\mu \left(\right\{r\in J\mid f\left(r\right)\notin V\left\}\right)$. If $g\in O(U,\delta )$, then$$\{r\in J\mid f(r)\oplus g(r)\notin V\}\subseteq \{r\in J\mid f(r)\notin V\}\cup \{r\in J\mid g(r)\notin U\}.$$It follows that$$\begin{array}{cc}\hfill \mu \left(\right\{r\in J\mid f\left(r\right)\oplus g\left(r\right)\notin V\left\}\right)& \le \mu \left(\right\{r\in J\mid f\left(r\right)\notin V\left\}\right)+\mu \left(\right\{r\in J\mid g\left(r\right)\notin U\left\}\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& <\mu \left(\right\{r\in J\mid f\left(r\right)\notin V\left\}\right)+\delta \hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\u03f5.\hfill \end{array}$$Thus, $f\oplus g\in O(V,\u03f5)$. This proves that $f\oplus O(U,\delta )\subseteq O(V,\u03f5)$.
- (iv)
- Let $O(V,\u03f5)\in \mathcal{N}\left({e}^{\u2022}\right)$ and let $f,g\in {G}^{\u2022}$. Then, there is a partition $\{{a}_{0},{a}_{1},\cdots ,{a}_{n}\}$ of J such that both f and g are constant on each interval $[{a}_{k},{a}_{k+1})$. By item (iv) of Theorem 5, for each $k\in \{0,1,\cdots ,n-1\}$, there is a set ${U}_{k}\in \mathcal{N}\left(e\right)$ such that$$\begin{array}{cc}\hfill (f\left({a}_{k}\right)\oplus {U}_{k})\oplus g\left({a}_{k}\right)& \subseteq (f\left({a}_{k}\right)\oplus g\left({a}_{k}\right))\oplus V,\hfill \\ \hfill f\left({a}_{k}\right)\oplus ({U}_{k}\oplus g\left({a}_{k}\right))& \subseteq (f\left({a}_{k}\right)\oplus g\left({a}_{k}\right))\oplus V,\hfill \\ \hfill (f\left({a}_{k}\right)\oplus {U}_{k})\u229eg\left({a}_{k}\right)& \subseteq (f\left({a}_{k}\right)\u229eg\left({a}_{k}\right))\oplus V.\hfill \end{array}$$By item (v) of Theorem 5, there is a set $U\in \mathcal{N}\left(e\right)$ such that $U\subseteq \bigcap _{k=0}^{n-1}{U}_{k}$. Furthermore,$$\begin{array}{cc}\hfill \left(f\right(r)\oplus U)\oplus g\left(r\right)& \subseteq \left(f\right(r)\oplus g(r\left)\right)\oplus V,\hfill \\ \hfill f\left(r\right)\oplus (U\oplus g(r\left)\right)& \subseteq \left(f\right(r)\oplus g(r\left)\right)\oplus V,\hfill \\ \hfill \left(f\right(r)\oplus U)\u229eg\left(r\right)& \subseteq \left(f\right(r)\u229eg(r\left)\right)\oplus V\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \{r\in J\mid (\ominus (f\oplus g)\oplus \left(\right(f\oplus h)\oplus g)\left)\right(r)\notin V\}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\{r\in J\mid \ominus (f\left(r\right)\oplus g\left(r\right))\oplus (\left(f\right(r)\oplus h(r\left)\right)\oplus g\left(r\right))\notin V\}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\{r\in J\mid (f\left(r\right)\oplus h\left(r\right))\oplus g(r)\notin (f\left(r\right)\oplus g\left(r\right))\oplus V\}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \subseteq \{r\in J\mid h(r)\notin U\}.\hfill \end{array}$$Hence, $\mu \left(\right\{r\in J\mid (\ominus (f\oplus g)\oplus ((f\oplus h)\oplus g\left)\right)\left(r\right)\notin V\left\}\right)\le \mu \left(\right\{r\in J\mid h\left(r\right)\notin U\left\}\right)<\u03f5$. It follows that $\ominus (f\oplus g)\oplus \left(\right(f\oplus h)\oplus g)\in O(V,\u03f5)$, and so $(f\oplus h)\oplus g\in (f\oplus g)\oplus O(V,\u03f5)$. This shows that $(f\oplus O(U,\u03f5\left)\right)\oplus g\subseteq (f\oplus g)\oplus O(V,\u03f5)$. Similarly, one can show that $f\oplus \left(O\right(U,\u03f5)\oplus g)\subseteq (f\oplus g)\oplus O(V,\u03f5)$ and that $(f\oplus O(U,\u03f5\left)\right)\u229eg\subseteq (f\u229eg)\oplus O(V,\u03f5)$.
- (v)
- Let $U,V\in \mathcal{N}\left(e\right)$, let $f\in {G}^{\u2022}$, and let $\u03f5,\delta >0$. Note that if $U\subseteq V$, then$$\{r\in J\mid f(r)\notin V\}\subseteq \{r\in J\mid f(r)\notin U\}.$$Therefore, $O(U,\u03f5)\subseteq O(V,\u03f5)$. Note that if $\u03f5\le \delta $, then $\mu \left(\right\{r\in J\mid f\left(r\right)\notin U\left\}\right)<\u03f5$ implies $\mu \left(\right\{r\in J\mid f\left(r\right)\notin U\left\}\right)<\delta $. This shows that $O(U,\u03f5)\subseteq O(U,\delta )$. Let $O(U,\u03f5),O(V,\delta )\in \mathcal{N}\left({e}^{\u2022}\right)$. By item (v) of Theorem 5, there is a set $W\in \mathcal{N}\left(e\right)$ such that $W\subseteq U\cap V$. Put ${\u03f5}_{0}=\mathrm{min}\{\u03f5,\delta \}$. The arguments above show that $O(W,{\u03f5}_{0})\subseteq O(U,\u03f5)\cap O(V,\delta )$.
- (vi)
- Clearly, ${e}^{\u2022}\in O(V,\u03f5)$ for all $V\in \mathcal{N}\left(e\right),\u03f5>0$. Let ${e}^{\u2022}\ne f\in {G}^{\u2022}$. Then, there exists a subinterval $[a,b)\subseteq J$ such that f is constant on $[a,b)$ and $f\left(a\right)\ne e$. By item (vi) of Theorem 5, there is a set $U\in \mathcal{N}\left(e\right)$ such that $f\left(a\right)\notin U$. Then, $[a,b)\subseteq \{r\in J\mid f(r)\notin U\}$. It follows that $\mu \left(\right\{r\in J\mid f\left(r\right)\notin U\left\}\right)\ge b-a$. Thus, $f\notin O(U,b-a)$. This shows that $\bigcap \mathcal{N}\left({e}^{\u2022}\right)=\left\{{e}^{\u2022}\right\}$.
- (vii)
- Let $O(V,\u03f5)\in \mathcal{N}\left({e}^{\u2022}\right)$, let $f,g\in {G}^{\u2022}$, and let $h\in O(V,\u03f5)$. It follows from the assumption that$$\begin{array}{cc}\hfill \{r\in J\mid \mathrm{gyr}[f,g\left]\right(h\left)\right(r)\notin V\}& =\{r\in J\mid \mathrm{gyr}[f\left(r\right),g\left(r\right)\left]\right(h\left(r\right))\notin V\}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \subseteq \{r\in J\mid h(r)\notin V\}.\hfill \end{array}$$Thus, $\mu \left(\right\{r\in J\mid \mathrm{gyr}[f,g]\left(h\right)\left(r\right)\notin V\left\}\right)\le \mu \left(\right\{r\in J\mid h\left(r\right)\notin V\left\}\right)<\u03f5$, which implies $\mathrm{gyr}[f,g]\left(h\right)\in O(V,\u03f5)$. This proves that $\mathrm{gyr}[f,g]\left(O\right(V,\u03f5\left)\right)\subseteq O(V,\u03f5)$.

**Remark**

**1.**

**Proposition**

**4.**

**Proof.**

**Theorem**

**8.**

**Proof.**

**Theorem**

**9.**

**Proof.**

**Theorem**

**10.**

**Proof.**

**Proposition**

**5.**

**Proof.**

**Theorem**

**11.**

- (i)
- if d is continuous, then so is ${d}^{\u2022}$;
- (ii)
- if d is a metric generating the topology of G, then ${d}^{\u2022}$ also generates the topology of ${G}^{\u2022}$.

**Proof.**

**Example**

**1.**

**Theorem**

**12.**

**Proof.**

**Theorem**

**13.**

**Proof.**

## 5. An Application to Normed Gyrogroups

**Definition**

**8**

**.**A real-valued function $\parallel \xb7\parallel $ on a gyrogroup G is called a gyronorm if it satisfies the following properties:

- 1.
- $\parallel x\parallel \ge 0$ for all $x\in G$ and $\parallel x\parallel =0$ if and only if $x=e$;
- 2.
- $\parallel \ominus x\parallel =\parallel x\parallel $ for all $x\in G$;
- 3.
- $\parallel x\oplus y\parallel \le \parallel x\parallel +\parallel y\parallel $ for all $x,y\in G$;
- 4.
- $\parallel \mathrm{gyr}[a,b]x\parallel =\parallel x\parallel $ for all $a,b,x\in G$.

**Proposition**

**6.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Proposition**

**7.**

**Proof.**

**Remark**

**2.**

**Proposition**

**8.**

**Proof.**

**Theorem**

**14.**

**Proof.**

**Proposition**

**9.**

**Proof.**

**Lemma**

**1.**

**Proof.**

**Theorem**

**15.**

**Proof.**

- (i)
- Let $r>0$. If $x,y\in B(e,\frac{r}{2})$, then$${d}_{\eta}(e,x\oplus y)={d}_{\eta}(\ominus x,y)\le {d}_{\eta}(\ominus x,e)+{d}_{\eta}(e,y)={d}_{\eta}(e,x)+{d}_{\eta}(e,y)<r.$$Thus, $B(e,\frac{r}{2})\oplus B(e,\frac{r}{2})\subseteq B(e,r)$.
- (ii)
- Let $r>0$ and let $x\in G$. By assumption, the function ${L}_{x}\circ \ominus \circ {R}_{x}$ is continuous and there is a number $\delta >0$ such that $({L}_{x}\circ \ominus \circ {R}_{x})\left(B(e,\delta )\right)\subseteq B(e,r)$. Thus, $x\oplus (\ominus (B(e,\delta )\oplus x\left)\right)\subseteq B(e,r)$.
- (iii)
- Let $r>0$ and let $x\in B(e,r)$. Put $\u03f5=r-{d}_{\eta}(e,x)$. If $y\in B(e,\u03f5)$, then$${d}_{\eta}(x\oplus y,e)=\parallel x\oplus y\parallel \le \parallel x\parallel +\parallel y\parallel <r.$$Therefore, $x\oplus B(e,\u03f5)\subseteq B(e,r)$.
- (iv)
- Let $r>0$ and let $x,y\in G$. By Proposition 7, the function ${R}_{\ominus y}^{-1}$, given by $\left({R}_{\ominus y}^{-1}\right)\left(z\right)=z\u229ey$ for all $z\in G$, is continuous. Note that $({R}_{\ominus y}^{-1}\circ {L}_{x})\left(e\right)=x\u229ey$. Since ${R}_{\ominus y}^{-1}\circ {L}_{x}$ is continuous, there is a number ${\delta}_{1}>0$ such that$$(x\oplus B(e,{\delta}_{1}))\u229ey=({R}_{\ominus y}^{-1}\circ {L}_{x})\left(B(e,{\delta}_{1})\right)\subseteq B(x\u229ey,r)=(x\u229ey)\oplus B(e,r).$$Similarly, there are numbers ${\delta}_{2},{\delta}_{3}>0$ such that $(x\oplus B(e,{\delta}_{2}))\oplus y\subseteq (x\oplus y)\oplus B(e,r)$ and $x\oplus (B(e,{\delta}_{3})\oplus y)\subseteq (x\oplus y)\oplus B(e,r)$. Put $\delta =\mathrm{min}\{{\delta}_{1},{\delta}_{2},{\delta}_{3}\}$. Then, the ball $B(e,\delta )$ does the job.
- (v)
- Let ${r}_{1},{r}_{2}>0$. Then, $B(e,r)\subseteq B(e,{r}_{1})\cap B(e,{r}_{2})$, where $r=\mathrm{min}\{{r}_{1},{r}_{2}\}$.
- (vi)
- Let $x\in G$. Then, $x\in {\displaystyle \bigcap _{r>0}}B(e,r)\iff {d}_{\eta}(e,x)=0\iff x=e$.
- (vii)
- Let $r>0$ and let $x,y\in G$. If $z\in B(e,r)$, then$${d}_{\eta}(e,\mathrm{gyr}[x,y]z)=\parallel \mathrm{gyr}[x,y]z\parallel =\parallel z\parallel ={d}_{\eta}(e,z)<r.$$Thus, $\mathrm{gyr}[x,y]\left(B\right(e,r\left)\right)\subseteq B(e,r)$.

**Definition**

**9.**

**Theorem**

**16.**

- (i)
- $(G,{\mathcal{T}}_{\eta})$ is a quasitopological gyrogroup;
- (ii)
- $(G,{\mathcal{T}}_{\eta})$ is a topological gyrogroup;
- (iii)
- $(G,{\mathcal{T}}_{\eta})$ is a strongly topological gyrogroup.

**Proof.**

## Author Contributions

## Funding

## Conflicts of Interest

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Wattanapan, J.; Atiponrat, W.; Suksumran, T.
Embedding of Strongly Topological Gyrogroups in Path-Connected and Locally Path-Connected Gyrogroups. *Symmetry* **2020**, *12*, 1817.
https://doi.org/10.3390/sym12111817

**AMA Style**

Wattanapan J, Atiponrat W, Suksumran T.
Embedding of Strongly Topological Gyrogroups in Path-Connected and Locally Path-Connected Gyrogroups. *Symmetry*. 2020; 12(11):1817.
https://doi.org/10.3390/sym12111817

**Chicago/Turabian Style**

Wattanapan, Jaturon, Watchareepan Atiponrat, and Teerapong Suksumran.
2020. "Embedding of Strongly Topological Gyrogroups in Path-Connected and Locally Path-Connected Gyrogroups" *Symmetry* 12, no. 11: 1817.
https://doi.org/10.3390/sym12111817