Let ⟨G,X,α⟩ be a G-space, where G is a non-Archimedean (having a local base at the identity consisting of open subgroups) and second countable topological group, and X is a zero-dimensional compact metrizable space. Let $⟨H\left({0,1}^{\aleph ₀}\right),{0,1}^{\aleph ₀},\tau ⟩$ be the natural (evaluation) action of the full group of autohomeomorphisms of the Cantor cube. Then (1) there exists a topological group embedding $\phi :G\hookrightarrow H\left({0,1}^{\aleph ₀}\right)$; (2) there exists an embedding $\psi :X\hookrightarrow {0,1}^{\aleph ₀}$, equivariant with respect to φ, such that ψ(X) is an equivariant retract of ${0,1}^{\aleph ₀}$ with respect to φ...

We investigate properties of coset topologies on commutative domains with an identity, in particular, the 𝓢-coprime topologies defined by Marko and Porubský (2012) and akin to the topology defined by Furstenberg (1955) in his proof of the infinitude of rational primes. We extend results about the infinitude of prime or maximal ideals related to the Dirichlet theorem on the infinitude of primes from Knopfmacher and Porubský (1997), and correct some results from that paper. Then we determine cluster...