Jan van Mill

We show that under the Continuum Hypothesis, the topological group of all homeomorphisms of the Čech–Stone remainder of $\omega $ with the $G_\delta $-topology is a universal object for all $P$-groups of weight at most $\mathfrak c$.

A well-known theorem of Noble states that each Tychonoff space X is homeomorphic to a closed subspace of a pseudocompact $$k_\mathbb {R}$$ k R -space. We strengthen this result by showing that any Tychonoff space X is homeomorphic to a closed subspace of an abelian pseudocompact $$k_\mathbb {R}$$ k R -group G such that $$w(G)\le \aleph _1\cdot w(X)$$ w ( G ) ≤ ℵ 1 · w ( X ) , and if, in addition, X is a precompact group, then X is topologically isomorphic to a closed subgroup of G . It is constructed the first examples of pseudocompact groups $$G_1$$ G 1 and $$G_2$$ G 2 (in fact, they are even countably compact and of weight $$\aleph _2$$ ℵ 2 ) such that $$G_1$$ G 1 is Ascoli but not a $$k_\mathbb {R}$$ k R -space, and $$G_2$$ G 2 is a $$k_\mathbb {R}$$ k R -space but not a k -space. Under $$\mathrm {MA+\lnot CH}$$ MA + ¬ CH , we show that any pseudocompact group of weight $$\aleph _1$$ ℵ 1 is Ascoli. These results are proved using topological properties of pseudocompact spaces X of weight $$\aleph _1$$ ℵ 1 and of $$\Sigma $$ Σ -products in products of compact spaces. Being motivated by these results and the countably compact part of Noble’s theorem, it is shown by a well-known technique that each countably compact infinite group has a separable countably compact subgroup of cardinality continuum.

We show that there are locally compact spaces that can be condensed on separable spaces, but not on compact separable spaces. We also show that for every cardinal $\kappa ,$ there is a locally compact topological group of cardinality $2^\kappa $ that can be condensed on a compact space but not on a compact topological group. These answer some questions of Arhangel’skii and Buzyakova.

We show that there are locally compact spaces that can be condensed onto separable spaces but not onto compact separable spaces. We also show that for every cardinal $κ$ there is a locally compact topological group of cardinality $2^κ$ that can be condensed onto a compact space but not onto a compact topological group. These answer some questions of Arhangel'skii and Buzyakova.

We show that under the Continuum Hypothesis, the topological group of all homeomorphisms of the Čech-Stone remainder of $ω$ with the $G_δ$-topology, is a universal object for all $P$-groups of weight at most ${\mathfrak c}$.

A topological space $X$ is a $Δ$-space (or $X \in Δ$) if for any decreasing sequence $\{A_n : n < ω\}$ of subsets of $X$ with empty intersection there is a (decreasing) sequence $\{U_n : n < ω\}$ of open sets with empty intersection such that $A_n \subset U_n$ for all $n < ω$. In this note we prove the following results concerning $Δ$-spaces. 1) Every $T_3$ countably compact $Δ$-space is compact. 2) If there is a $T_1$ crowded Baire $Δ$-space then there is an inner model with a measurable cardinal. 3) If $X \in Δ$ and $cf \big(o(X) \big) > ω$ then $|X| < o(X)$. (Here $o(X)$ is the number of open subsets of $X$.) The first two of these provide full and/or partial solutions to problems raised in the literature, while the third improves a known result.

Answering a question raised by V.V. Tkachuk in [10], we present several examples of $$\sigma$$ σ -compact spaces, some only consistent and some in ZFC, that are not countably tight but in which the closure of any discrete subset is countably tight. In fact, in some of our examples the closures of all discrete subsets are even first countable.