Some results in generalized topological groups

It is proved in this paper that if $G$ is a $\mathcal{G}$-topological group and $\{K_i:i\in \omega \}$ is a family of generalized open subsets containing the identity element $e$ in $G$ satisfying $K_{i+1}^2\subset K_i$ and $K_i^{-1}=K_i$ for every $i\in \omega$, then $G/H$ is metrizable where $H=\bigcap _{i\in \omega }{K}_i$. Let $\left(G,\tau \right)$ be a $\mathcal{G}$-topological group satisfying for every $e\in U\in \tau$ that there is $e\in O\in \tau$ such that $O^2\subset U$. Then, we obtain that 1) if $H$ is generalized $\kappa$-narrow and $c{l}_{G}H=G$, then $G$ is also generalized $\kappa$-narrow; 2) $AB$ is generalized $\kappa$-narrow in $G$ provided that $A$ and $B$ are generalized $\kappa$-narrow in $G$. Finally, we consider coset spaces. It is shown that 3) for a subgroup $H$ of G, if H and the generalized quotient space G/H have countable generalized pseudocharacter, then G also has countable generalized pseudocharacter; 4) for a subgroup $H$, if $H$ is the generalized connected component containing $e$ in $G$, $G/H$ is generalized totally disconnected; 5) if H is a generalized closed subgroup and generalized totally disconnected, then G is also generalized totally disconnected when G/H is generalized totally disconnected.