It is established that a remainder of a non-locally compact topological group $G$ has the Baire property if and only if the space $G$ is not Čech-complete. We also show that if $G$ is a non-locally compact topological group of countable tightness, then either $G$ is submetrizable, or $G$ is the Čech-Stone remainder of an arbitrary remainder $Y$ of $G$. It follows that if $G$ and $H$ are non-submetrizable topological groups of countable tightness such that some remainders of $G$ and $H$ are homeomorphic, then the spaces...

The authors prove the following result, which generalizes a well-known theorem of I. Glicksberg [G]: If G is a locally compact Abelian group with Bohr compactification bG, and if N is a closed metrizable subgroup of bG, then every A ⊆ G satisfies: A·(N ∩ G) is compact in G if and only if {aN:a ∈ A} is compact in bG/N. Examples are given to show: (a) the asserted equivalence can fail in the absence of the metrizability hypothesis, even when N ∩ G = {1}; (b) the asserted equivalence can hold for suitable...

Let G be a maximally almost periodic (MAP) Abelian group and let ℬ be a boundedness on G in the sense of Vilenkin. We study the relations between ℬ and the Bohr topology of G for some well known groups with boundedness (G,ℬ). As an application, we prove that the Bohr topology of a topological group which is topologically isomorphic to the direct product of a locally convex space and an ${ℒ}_{\infty }$-group, contains “many” discrete C-embedded subsets which are C*-embedded in their Bohr compactification. This...

Throughout this abstract, $G$ is a topological Abelian group and $\widehat{G}$ is the space of continuous homomorphisms from $G$ into the circle group $𝕋$ in the compact-open topology. A dense subgroup $D$ of $G$ is said to determine $G$ if the (necessarily continuous) surjective isomorphism $\widehat{G}\twoheadrightarrow \widehat{D}$ given by $h\mapsto h|D$ is a homeomorphism, and $G$ is determined if each dense subgroup of $G$ determines $G$. The principal result in this area, obtained independently by L. Außenhofer and M. J. Chasco, is the following: Every metrizable group is...

For any topological group $G$ the dual object $\widehat{G}$ is defined as the set of equivalence classes of irreducible unitary representations of $G$ equipped with the Fell topology. If $G$ is compact, $\widehat{G}$ is discrete. In an earlier paper we proved that $\widehat{G}$ is discrete for every metrizable precompact group, i.e. a dense subgroup of a compact metrizable group. We generalize this result to the case when $G$ is an almost metrizable precompact group.

We show that there exist ${\omega }_{\mu }$-metrizable spaces which do not have the Dugundji extension property ($2^{{\omega }_1}$ with the countable box topology is such a space). This answers a question posed by the second author in 1972, and shows that certain results of van Douwen and Borges are false.

It was known that free Abelian groups do not admit a Hausdorff compact group topology. Tkachenko showed in 1990 that, under CH, a free Abelian group of size $ℭ$ admits a Hausdorff countably compact group topology. We show that no Hausdorff group topology on a free Abelian group makes its $\omega$-th power countably compact. In particular, a free Abelian group does not admit a Hausdorff $p$-compact nor a sequentially compact group topology. Under CH, we show that a free Abelian group does not admit a Hausdorff...

An embedding X ⊂ G of a topological space X into a topological group G is called functorial if every homeomorphism of X extends to a continuous group homomorphism of G. It is shown that the interval [0, 1] admits no functorial embedding into a finite-dimensional or metrizable topological group.

We introduce and study, following Z. Frol’ık, the class $ℬ\left(𝒫\right)$ of regular $P$-spaces $X$ such that the product $X\times Y$ is pseudo-${\aleph }_1$-compact, for every regular pseudo-${\aleph }_1$-compact $P$-space $Y$. We show that every pseudo-${\aleph }_1$-compact space which is locally $ℬ\left(𝒫\right)$ is in $ℬ\left(𝒫\right)$ and that every regular Lindelöf $P$-space belongs to $ℬ\left(𝒫\right)$. It is also proved that all pseudo-${\aleph }_1$-compact $P$-groups are in $ℬ\left(𝒫\right)$. The problem of characterization of subgroups of $ℝ$-factorizable (equivalently, pseudo-${\aleph }_1$-compact) $P$-groups is considered as well. We give some necessary...

For every discrete group $G$, the Stone-Čech compactification $\beta G$ of $G$ has a natural structure of a compact right topological semigroup. An ultrafilter $p\in G^{*}$, where $G^{*}=\beta G\setminus G$, is called right cancellable if, given any $q,r\in G^{*}$, $qp=rp$ implies $q=r$. For every right cancellable ultrafilter $p\in G^{*}$, we denote by $G\left(p\right)$ the group $G$ endowed with the strongest left invariant topology in which $p$ converges to the identity of $G$. For any countable group $G$ and any right cancellable ultrafilters $p,q\in G^{*}$, we show that $G\left(p\right)$ is homeomorphic to $G\left(q\right)$ if and only if...

Using classical results of infinite-dimensional geometry, we show that the isometry group of the Urysohn space, endowed with its usual Polish group topology, is homeomorphic to the separable Hilbert space ℓ²(ℕ). The proof is based on a lemma about extensions of metric spaces by finite metric spaces, which we also use to investigate (answering a question of I. Goldbring) the relationship, when A,B are finite subsets of the Urysohn space, between the group of isometries fixing A pointwise, the group...

We consider the spaces called $Seq\left(u_t\right)$, constructed on the set $Seq$ of all finite sequences of natural numbers using ultrafilters $u_t$ to define the topology. For such spaces, we discuss continuity, homogeneity, and rigidity. We prove that $S\left(u_t\right)$ is homogeneous if and only if all the ultrafilters $u_t$ have the same Rudin-Keisler type. We proved that a space of Louveau, and in certain cases, a space of Sirota, are homeomorphic to $Seq\left(p\right)$ (i.e., $u_t=p$ for all $t\in Seq$). It follows that for a Ramsey ultrafilter $p$, $Seq\left(p\right)$ is a topological group....

We prove a Dichotomy Theorem: for each Hausdorff compactification $bG$ of an arbitrary topological group $G$, the remainder $bG\setminus G$ is either pseudocompact or Lindelöf. It follows that if a remainder of a topological group is paracompact or Dieudonne complete, then the remainder is Lindelöf, and the group is a paracompact $p$-space. This answers a question in A.V. Arhangel’skii, Some connections between properties of topological groups and of their remainders, Moscow Univ. Math. Bull. 54:3 (1999), 1–6. It is...