We show that every subgroup of an $ℝ$-factorizable abelian $P$-group is topologically isomorphic to a closed subgroup of another $ℝ$-factorizable abelian $P$-group. This implies that closed subgroups of $ℝ$-factorizable $P$-groups are not necessarily $ℝ$-factorizable. We also prove that if a Hausdorff space $Y$ of countable pseudocharacter is a continuous image of a product $X={\prod }_{i\in I}{X}_i$ of $P$-spaces and the space $X$ is pseudo-${\omega }_1$-compact, then $nw\left(Y\right)\le {\aleph }_0$. In particular, direct products of $ℝ$-factorizable $P$-groups are $ℝ$-factorizable and...

The properties of $ℝ$-factorizable groups and their subgroups are studied. We show that a locally compact group $G$ is $ℝ$-factorizable if and only if $G$ is $\sigma$-compact. It is proved that a subgroup $H$ of an $ℝ$-factorizable group $G$ is $ℝ$-factorizable if and only if $H$ is $z$-embedded in $G$. Therefore, a subgroup of an $ℝ$-factorizable group need not be $ℝ$-factorizable, and we present a method for constructing non-$ℝ$-factorizable dense subgroups of a special class of $ℝ$-factorizable groups. Finally, we construct a closed...

We present several sum theorems for Ohio completeness. We prove that Ohio completeness is preserved by taking σ-locally finite closed sums and also by taking point-finite open sums. We provide counterexamples to show that Ohio completeness is preserved neither by taking locally countable closed sums nor by taking countable open sums.

We construct a consistent example of a normal locally compact metacompact space which is not paracompact, answering a question of A. V. Arkhangel’skiĭ and F. Tall. An interplay between a tower in P(ω)/Fin, an almost disjoint family in ${\left[\omega \right]}^{\omega }$, and a version of an (ω,1)-morass forms the core of the proof. A part of the poset which forces the counterexample can be considered a modification of a poset due to Judah and Shelah for obtaining a Q-set by a countable support iteration.

For a separable metric space X, we consider possibilities for the sequence $S\left(X\right)=d_n:n\in \mathbb{N}$ where $d_n=dim{X}^n$. In Section 1, a general method for producing examples is given which can be used to realize many of the possible sequences. For example, there is $X_n$ such that $S\left(X_n\right)=n,n+1,n+2,...$, $Y_n$, for n >1, such that $S\left(Y_n\right)=n,n+1,n+2,n+2,n+2,...$, and Z such that S(Z) = 4, 4, 6, 6, 7, 8, 9,.... In Section 2, a subset X of ${ℝ}^2$ is shown to exist which satisfies $1=dimX=dim{X}^2$ and $dim{X}^3=2$.

Two Boolean algebras are elementarily equivalent if and only if they satisfy the same first-order statements in the language of Boolean algebras. We prove that every Boolean algebra is elementarily equivalent to the algebra of clopen subsets of a normal P-space.

We give an example of an extremally disconnected compact Hausdorff space with an open continuous selfmap such that the fixed point set is nonvoid and nowhere dense, respṫhat there is exactly one nonisolated fixed point.

The Golomb space ${\mathbb{N}}_{\tau }$ is the set $\mathbb{N}$ of positive integers endowed with the topology $\tau$ generated by the base consisting of arithmetic progressions $\{a+bn:n\ge 0\}$ with coprime $a,b$. We prove that the Golomb space ${\mathbb{N}}_{\tau }$ is topologically rigid in the sense that its homeomorphism group is trivial. This resolves a problem posed by T. Banakh at Mathoverflow in 2017.

The Open Colouring Axiom implies that the measure algebra cannot be embedded into P(ℕ)/fin. We also discuss errors in previous results on the embeddability of the measure algebra.

We prove that the Niemytzki plane is $\varkappa$-metrizable and we try to explain the differences between the concepts of a stratifiable space and a $\varkappa$-metrizable space. Also, we give a characterisation of $\varkappa$-metrizable spaces which is modelled on the version described by Chigogidze.

Two compact spaces are co-absoluteif their respective regular open algebras are isomorphic (i.e. homeomorphic Gleason covers). We prove that it is consistent that βω and βℝ are not co-absolute.

Hewitt [Rings of real-valued continuous functions. I., Trans. Amer. Math. Soc. 64 (1948), 45–99] defined the $m$-topology on $C\left(X\right)$, denoted $C_m\left(X\right)$, and demonstrated that certain topological properties of $X$ could be characterized by certain topological properties of $C_m\left(X\right)$. For example, he showed that $X$ is pseudocompact if and only if $C_m\left(X\right)$ is a metrizable space; in this case the $m$-topology is precisely the topology of uniform convergence. What is interesting with regards to the $m$-topology is that it is possible, with...