Some results in generalized topological groups
Kaixiong Lin , Jing Zhang , Hanfeng Wang
Electronic Research Archive ›› 2025, Vol. 33 ›› Issue (10) : 6206 -6218.
It is proved in this paper that if G is a $\mathcal{G}$-topological group and {Ki : i ∈ ω} is a family of generalized open subsets containing the identity element e in G satisfying K2i+1 ⊂ Ki and K-1i = Ki for every i ∈ ω, then G/H is metrizable where H = ∩i∈ω Ki. Let (G, τ) be a G-topological group satisfying for every e ∈ U ∈ τ that there is e ∈ O ∈ τ such that O2 ⊂ U. Then, we obtain that 1) if H is generalized κ-narrow and clG H = G, then G is also generalized κ-narrow; 2) AB is generalized κ-narrow in G provided that A and B are generalized κ-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.
generalized topology groups / coset spaces / total disconnectedness / cardinal invariants / metrizability
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