Let (G,τ) be a Hausdorff Abelian topological group. It is called an s-group (resp. a bs-group) if there is a set S of sequences in G such that τ is the finest Hausdorff (resp. precompact) group topology on G in which every sequence of S converges to zero. Characterizations of Abelian s- and bs-groups are given. If (G,τ) is a maximally almost periodic (MAP) Abelian s-group, then its Pontryagin dual group ${(G,\tau )}^{\wedge }$ is a dense -closed subgroup of the compact group ${\left(G_d\right)}^{\wedge }$, where $G_d$ is the group G with the discrete...

A well known theorem of W.W. Comfort and K.A. Ross, stating that every pseudocompact group is $C$-embedded in its Weil completion [5] (which is a compact group), is extended to some new classes of topological groups, and the proofs are very transparent, short and elementary (the key role in the proofs belongs to Lemmas 1.1 and 4.1). In particular, we introduce a new notion of canonical uniform tightness of a topological group $G$ and prove that every $G_{\delta }$-dense subspace $Y$ of a topological group $G$, such...

We continue the study of topological properties of the group Homeo(X) of all homeomorphisms of a Cantor set X with respect to the uniform topology τ, which was started by Bezuglyi, Dooley, Kwiatkowski and Medynets. We prove that the set of periodic homeomorphisms is τ-dense in Homeo(X) and deduce from this result that the topological group (Homeo(X),τ) has the Rokhlin property, i.e., there exists a homeomorphism whose conjugacy class is τ-dense in Homeo(X). We also show that for any homeomorphism...

Let $X$ be an Abelian topological group. A subgroup $H$ of $X$ is characterized if there is a sequence $𝐮=\left\{u_n\right\}$ in the dual group of $X$ such that $H=\{x\in X:(u_n,x)\to 1\}$. We reduce the study of characterized subgroups of $X$ to the study of characterized subgroups of compact metrizable Abelian groups. Let $c_0\left(X\right)$ be the group of all $X$-valued null sequences and ${𝔲}_0$ be the uniform topology on $c_0\left(X\right)$. If $X$ is compact we prove that $c_0\left(X\right)$ is a characterized subgroup of $X^{\mathbb{N}}$ if and only if $X\cong {𝕋}^n\times F$, where $n\ge 0$ and $F$ is a finite Abelian group. For every compact Abelian...

Let G be a locally compact group, and let U be its unitary representation on a Hilbert space H. Endow the space ℒ(H) of bounded linear operators on H with the weak operator topology. We prove that if U is a measurable map from G to ℒ(H) then it is continuous. This result was known before for separable H. We also prove that the following statement is consistent with ZFC: every measurable homomorphism from a locally compact group into any topological group is continuous.

We prove that it is independent of ZFC whether every Hausdorff countable space of weight less than $c$ has a dense regular subspace. Examples are given of countable Hausdorff spaces of weight $c$ which do not have dense Urysohn subspaces. We also construct an example of a countable Urysohn space, which has no dense completely Hausdorff subspace. On the other hand, we establish that every Hausdorff space of $\pi$-weight less than $𝔭$ has a dense completely Hausdorff (and hence Urysohn) subspace. We show that...

The convolution of ultrafilters of closed subsets of a normal topological group is considered as a substitute of the extension onto ${\left(\beta \right)}^2$ of the group operation. We find a subclass of ultrafilters for which this extension is well-defined and give some examples of pathologies. Next, for a given locally compact group and its dense subgroup , we construct subsets of β algebraically isomorphic to . Finally, we check whether the natural mapping from β onto β is a homomorphism with respect to the extension...

We will show that under ${MA}_{countable}$ for each $k\in \mathbb{N}$ there exists a group whose $k$-th power is countably compact but whose $2^k$-th power is not countably compact. In particular, for each $k\in \mathbb{N}$ there exists $l\in [k,2^k)$ and a group whose $l$-th power is countably compact but the $l+1$-st power is not countably compact.

We investigate hereditarily normal topological groups and their subspaces. We prove that every compact subspace of a hereditarily normal topological group is metrizable. To prove this statement we first show that a hereditarily normal topological group with a non-trivial convergent sequence has $G_{\delta }$-diagonal. This implies, in particular, that every countably compact subspace of a hereditarily normal topological group with a non-trivial convergent sequence is metrizable. Another corollary is that under...

Let $(G,\tau )$ be a commutative Hausdorff locally solid lattice group. In this paper we prove the following: (1) If $(G,\tau )$ has the $A$(iii)-property, then its completion $(\widehat{G},\widehat{\tau })$ is an order-complete locally solid lattice group. (2) If $G$ is order-complete and $\tau$ has the Fatou property, then the order intervals of $G$ are $\tau$-complete. (3) If $(G,\tau )$ has the Fatou property, then $G$ is order-dense in $\widehat{G}$ and $(\widehat{G},\widehat{\tau })$ has the Fatou property. (4) The order-bound topology on any commutative lattice group is the finest locally solid topology on...

It is proved that every uncountable $\omega$-bounded group and every homogeneous space containing a convergent sequence are resolvable. We find some conditions for a topological group topology to be irresolvable and maximal.

The main purpose of this paper is to show that any localic group is complete in its two-sided uniformity, settling a problem open since work began in this area a decade ago. In addition, a number of other results are established, providing in particular a new functor from topological to localic groups and an alternative characterization of $LT$-groups.

For a topological property P, we say that a space X is star Pif for every open cover Uof the space X there exists Y ⊂ X such that St(Y,U) = X and Y has P. We consider star countable and star Lindelöf spaces establishing, among other things, that there exists first countable pseudocompact spaces which are not star Lindelöf. We also describe some classes of spaces in which star countability is equivalent to countable extent and show that a star countable space with a dense σ-compact subspace can have...

Minimal Hausdorff (Baire) group topologies of certain groups of transformations naturally occurring in analysis are studied. The results obtained are subsequently applied to show that, e.g., the homeomorphism groups of the rational and of the irrational numbers carry no Polish group topology. In answer to a question of A. S. Kechris it is shown that the group of Borel automorphisms of ℝ cannot be a Polish group either.