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 $T_{2}$.

We consider generalised metrisability and cardinal invariants in quasitopological groups. We construct examples to show that some equalities of cardinal invariants in topological groups cannot be extended to quasitopological groups.

We demonstrate how most common cardinal invariants associated with a von Neumann algebra $\mathcal{M}$ can be computed from the decomposability number, $\text{dens}\left( \mathcal{M} \right)$, and the minimal cardinality of a generating set, $\text{gen}\left( \mathcal{M} \right)$. Applications include the equivalence of the well-known generator problem, “Is every separably-acting von Neumann algebra singly-generated?”, with the formally stronger questions, “Is every countably-generated von Neumann algebra singly-generated?” and “Is the gen invariant monotone?” Modulo the generator problem, we determine the range of the invariant $\left( \text{gen}\left( \mathcal{M} \right),\,\text{dens}\left( \mathcal{M} \right) \right)$ , which is mostly governed by the inequality $\text{dens}\left( \mathcal{M} \right)\,\le {{\mathfrak{C}}^{\text{gen}\left( \mathcal{M} \right)}}$.