Entropy of measure-preserving or continuous actions of amenable discrete groups allows for various equivalent approaches. Among them are those given by the techniques developed by Ollagnier and Pinchon on the one hand and the Ornstein–Weiss lemma on the other. We extend these two approaches to the context of actions of amenable topological groups. In contrast to the discrete setting, our results reveal a remarkable difference between the two concepts of entropy in the realm of non-discrete groups: while the first quantity collapses to 0 in the non-discrete case, the second yields a well-behaved invariant for amenable unimodular groups. Concerning the latter, we moreover study the corresponding notion of topological pressure, prove a Goodwyn-type theorem, and establish the equivalence with the uniform lattice approach (for locally compact groups admitting a uniform lattice). Our study elaborates on a version of the Ornstein–Weiss lemma due to Gromov.

To study when a paratopological group becomes a topological group, Arhangel’skii et al. [‘Topological games and topologies on groups’, Math. Maced. 8 (2010), 1–19] introduced the class of $(\,\unicode[STIX]{x1D6FD},G_{\unicode[STIX]{x1D6F1}})$-unfavourable spaces. We show that every $\unicode[STIX]{x1D707}$-complete (or normal) $(\,\unicode[STIX]{x1D6FD},G_{\unicode[STIX]{x1D6F1}})$-unfavourable semitopological group is a topological group. We prove that the product of a $(\,\unicode[STIX]{x1D6FD},G_{\unicode[STIX]{x1D6F1}})$-unfavourable space and a strongly Fréchet $(\unicode[STIX]{x1D6FC},G_{\unicode[STIX]{x1D6F1}})$-favourable space is $(\,\unicode[STIX]{x1D6FD},G_{\unicode[STIX]{x1D6F1}})$-unfavourable. We also show that continuous closed irreducible mappings preserve the $(\,\unicode[STIX]{x1D6FD},G_{\unicode[STIX]{x1D6F1}})$-unfavourableness in both directions.

The Banach–Mazur separable quotient problem asks whether every infinite-dimensional Banach space $B$ has a quotient space that is an infinite-dimensional separable Banach space. The question has remained open for over 80 years, although an affirmative answer is known in special cases such as when $B$ is reflexive or even a dual of a Banach space. Very recently, it has been shown to be true for dual-like spaces. An analogous problem for topological groups is: Does every infinite-dimensional (in the topological sense) connected (Hausdorff) topological group $G$ have a quotient topological group that is infinite dimensional and metrisable? While this is known to be true if $G$ is the underlying topological group of an infinite-dimensional Banach space, it is shown here to be false even if $G$ is the underlying topological group of an infinite-dimensional locally convex space. Indeed, it is shown that the free topological vector space on any countably infinite $k_{\unicode[STIX]{x1D714}}$-space is an infinite-dimensional toplogical vector space which does not have any quotient topological group that is infinite dimensional and metrisable. By contrast, the Graev free abelian topological group and the Graev free topological group on any infinite connected Tychonoff space, both of which are connected topological groups, are shown here to have the tubby torus $\mathbb{T}^{\unicode[STIX]{x1D714}}$, which is an infinite-dimensional metrisable group, as a quotient group.

We investigate continuous transitive actions of semitopological groups on spaces, as well as separately continuous transitive actions of topological groups.

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 extend Følner’s amenability criterion to the realm of general topological groups. Building on this, we show that a topological group $G$ is amenable if and only if its left-translation action can be approximated in a uniform manner by amenable actions on the set  $G$ . As applications we obtain a topological version of Whyte’s geometric solution to the von Neumann problem and give an affirmative answer to a question posed by Rosendal.

Let the measure algebra of a topological group $G$ be equipped with the topology of uniform convergence on bounded right uniformly equicontinuous sets of functions. Convolution is separately continuous on the measure algebra, and it is jointly continuous if and only if $G$ has the $\text{SIN}$ property. On the larger space $\text{LUC}{{(G)}^{*}}$, which includes the measure algebra, convolution is also jointly continuous if and only if the group has the $\text{SIN}$ property, but not separately continuous for many non-SIN groups.

In this paper, it is shown that every compact Hausdorff $K$-space has countable tightness. This result gives a positive answer to a problem posed by Malykhin and Tironi [‘Weakly Fréchet–Urysohn and Pytkeev spaces’, Topology Appl.104 (2000), 181–190]. We show that a semitopological group $G$ that is a $K$-space is first countable if and only if $G$ is of point-countable type. It is proved that if a topological group $G$ is a $K$-space and has a locally paracompact remainder in some Hausdorff compactification, then $G$ is metrisable.

We present a structure theorem for a broad class of homeomorphisms of finite order on countable zero dimensional spaces. As applications we show the following.

1. (a) Every countable nondiscrete topological group not containing an open Boolean subgroup can be partitioned into infinitely many dense subsets.

2. (b) If $G$ is a countably infinite Abelian group with finitely many elements of order 2 and $\beta G$ is the Stone–Čech compactification of $G$ as a discrete semigroup, then for every idempotent $p\,\,\in \,\,\beta G\backslash \{0\}$, the subset $\{p,-p\}\subset \beta G$ generates algebraically the free product of one-element semigroups $\{p\}$ and $\{-p\}$.