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 $ cl_{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.