We prove a general theorem about preservation of the covering dimension $dim$ by certain covariant functors that implies, among others, the following concrete results. If $G$ G is a pathwise connected separable metric...

Let $\tau$ be an uncountable regular cardinal and $G$ a $T_1$ topological group. We prove the following statements: (1) If $\tau$ is homeomorphic to a closed subspace of $G$, $G$ is Abelian, and the order of every non-neutral element of $G$ is greater than $5$ then $\tau \times \tau$ embeds in $G$ as a closed subspace. (2) If $G$ is Abelian, algebraically generated by $\tau \subset G$, and the order of every element does not exceed $3$ then $\tau \times \tau$ is not embeddable in $G$. (3) There exists an Abelian topological group $H$ such that ${\omega }_1$ is homeomorphic to a closed subspace...

We show that there exists an Abelian topological group $G$ such that the operations in $G$ cannot be extended to the Dieudonné completion $\mu G$ of the space $G$ in such a way that $G$ becomes a topological subgroup of the topological group $\mu G$. This provides a complete answer to a question of V.G. Pestov and M.G. Tkačenko, dating back to 1985. We also identify new large classes of topological groups for which such an extension is possible. The technique developed also allows to find many new solutions to the...

The natural quotient map q from the space of based loops in the Hawaiian earring onto the fundamental group provides a naturally occuring example of a quotient map such that q × q fails to be a quotient map. With the quotient topology, this example shows π₁(X,p) can fail to be a topological group if X is locally path connected.

We study retracts of coset spaces. We prove that in certain spaces the set of points that are contained in a component of dimension less than or equal to n, is a closed set. Using our techniques we are able to provide new examples of homogeneous spaces that are not coset spaces. We provide an example of a compact homogeneous space which is not a coset space. We further provide an example of a compact metrizable space which is a retract of a homogeneous compact space, but which is not a retract of...

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...