A Common Extension of Arhangel’skĭ’s Theorem and the Hajnal–Juhász Inequality

We present a result about $G_{\unicode[STIX]{x1D6FF}}$ covers of a Hausdorff space that implies various known cardinal inequalities, including the following two fundamental results in the theory of cardinal invariants in topology: $|X|\leqslant 2^{L(X)\unicode[STIX]{x1D712}(X)}$ (Arhangel’skiĭ) and $|X|\leqslant 2^{c(X)\unicode[STIX]{x1D712}(X)}$ (Hajnal–Juhász). This solves a question that goes back to Bell, Ginsburg and Woods’s 1978 paper (M. Bell, J.N. Ginsburg and R.G. Woods, Cardinal inequalities for topological spaces involving the weak Lindelöf number, Pacific J. Math. 79(1978), 37–45) and is mentioned in Hodel’s survey on Arhangel’skiĭ’s Theorem (R. Hodel, Arhangel’skii’s solution to Alexandroff’s problem: A survey, Topology Appl. 153(2006), 2199–2217).In contrast to previous attempts, we do not need any separation axiom beyond $T_{2}$.

Cardinal invariants of paratopological groups

We show that a regular totally ω-narrow paratopological group G has countable index of regularity, i.e., for every neighborhood U of the identity e of G, we can find a neighborhood V of e and a countable family of neighborhoods of e in G such that ∩W∈γ VW−1⊆ U. We prove that every regular (Hausdorff) totally !-narrow paratopological group is completely regular (functionally Hausdorff). We show that the index of regularity of a regular paratopological group is less than or equal to the weak Lindelöf number. We also prove that every Hausdorff paratopological group with countable π- character has a regular Gσ-diagonal.