Quasitopological groups, covering properties and cardinal inequalities

Abstract

In this paper, we make several observations on quasitopological groups, covering properties and cardinal inequalities. In particular, we show that: (1) For every quasitopological group G, ib(G)≤e(G); (2) If G is a T1-quasitopological group then G has cardinality at most 2e(G)ψ(G), which partially answers a question asked by Arhangel'skii and Tkachenko in [3] and a question asked by Tkachenko in [20]; (3) In the class of first countable quasi-topological groups, dually separable is self-dual with respect to neighborhood assignments; (4) If X is a Baire weakly star countable space with a rank 2-diagonal then X has cardinality at most 2ω, which gives a partial answer to a question of Gotchev in [11]; (5) If X is a quasiLindelöf space with a rank 2-diagonal then X has cardinality at most 2ω; (6) In any model of ZFC containing a 2ω-Suslin Line, there exists a Moore space of cellularity at most 2ω which has cardinality greater than 2ω, which answers a question of Arhangel'skii and Bella in [2]. Some new questions are also posed.