We show that a regular totally ω-narrow paratopological group G has countable index of regularity, i.e., for every neighborhood U of the identity e of G, we can find a neighborhood V of e and a countable family of neighborhoods of e in G such that ∩W∈γ VW−1⊆ U. We prove that every regular (Hausdorff) totally !-narrow paratopological group is completely regular (functionally Hausdorff). We show that the index of regularity of a regular paratopological group is less than or equal to the weak Lindelöf...

We study relations between the cellularity and index of narrowness in topological groups and their $G_{\delta }$-modifications. We show, in particular, that the inequalities $in\left({\left(H\right)}_{\tau }\right)\le 2^{\tau ·in\left(H\right)}$ and $c\left({\left(H\right)}_{\tau }\right)\le 2^{2^{\tau ·in\left(H\right)}}$ hold for every topological group $H$ and every cardinal $\tau \ge \omega$, where ${\left(H\right)}_{\tau }$ denotes the underlying group $H$ endowed with the $G_{\tau }$-modification of the original topology of $H$ and $in\left(H\right)$ is the index of narrowness of the group $H$. Also, we find some bounds for the complexity of continuous real-valued functions $f$ on an arbitrary $\omega$-narrow group $G$ understood...

Given a discrete group $G$, we consider the set $ℒ\left(G\right)$ of all subgroups of $G$ endowed with topology of pointwise convergence arising from the standard embedding of $ℒ\left(G\right)$ into the Cantor cube ${\{0,1\}}^G$. We show that the cellularity $c\left(ℒ\left(G\right)\right)\le {\aleph }_0$ for every abelian group $G$, and, for every infinite cardinal $\tau$, we construct a group $G$ with $c\left(ℒ\right(G\left)\right)=\tau$.

We study homeomorphism groups of metrizable compactifications of ℕ. All of those groups can be represented as almost zero-dimensional Polishable subgroups of the group $S_{\infty }$. As a corollary, we show that all Polish groups are continuous homomorphic images of almost zero-dimensional Polishable subgroups of $S_{\infty }$. We prove a sufficient condition for these groups to be one-dimensional and also study their descriptive complexity. In the last section we associate with every Polishable ideal on ℕ a certain Polishable...

We present an example of a complete ${\aleph }_0$-bounded topological group $H$ which is not $ℝ$-factorizable. In addition, every $G_{\delta }$-set in the group $H$ is open, but $H$ is not Lindelöf.

Let $(L,𝒯)$ be a Tychonoff (regular) paratopological group or algebra over a field or ring $K$ or a topological semigroup. If $nw(L,𝒯)\le \tau$ and $nw\left(K\right)\le \tau$, then there exists a Tychonoff (regular) topology ${𝒯}^{*}\subseteq 𝒯$ such that $w(L,{𝒯}^{*})\le \tau$ and $(L,{𝒯}^{*})$ is a paratopological group, algebra over $K$ or a topological semigroup respectively.

We prove that every connected locally compact Abelian topological group is sequentially connected, i.e., it cannot be the union of two proper disjoint sequentially closed subsets. This fact is then applied to the study of extensions of topological groups. We show, in particular, that if $H$ is a connected locally compact Abelian subgroup of a Hausdorff topological group $G$ and the quotient space $G/H$ is sequentially connected, then so is $G$.

It is shown that both the free topological group $F\left(X\right)$ and the free Abelian topological group $A\left(X\right)$ on a connected locally connected space $X$ are locally connected. For the Graev’s modification of the groups $F\left(X\right)$ and $A\left(X\right)$, the corresponding result is more symmetric: the groups $F\Gamma \left(X\right)$ and $A\Gamma \left(X\right)$ are connected and locally connected if $X$ is. However, the free (Abelian) totally bounded group $FTB\left(X\right)$ (resp., $ATB\left(X\right)$) is not locally connected no matter how “good” a space $X$ is. The above results imply that every non-trivial continuous homomorphism...

A subset of a Polish space X is called universally small if it belongs to each ccc σ-ideal with Borel base on X. Under CH in each uncountable Abelian Polish group G we construct a universally small subset A₀ ⊂ G such that |A₀ ∩ gA₀| = for each g ∈ G. For each cardinal number κ ∈ [5,⁺] the set A₀ contains a universally small subset A of G with sharp packing index $pack♯\left(A_{\kappa }\right)=sup\left|\right|⁺:\subset {gA}_{g\in G}isdisjoint$ equal to κ.

In this paper, we show that it is possible to extend the Ellis theorem, establishing the relations between axioms of a topological group on a new class $𝒩$ of spaces containing all countably compact spaces in the case of Abelian group structure. We extend statements of the Ellis theorem concerning separate and joint continuity of group inverse on the class of spaces $𝒩$ that gives some new examples and statements for the $C_p$-theory and theory of topologically homogeneous spaces.

In 2008 Juhász and Szentmiklóssy established that for every compact space $X$ there exists a discrete $D\subset X\times X$ with $\left|D\right|=d\left(X\right)$. We generalize this result in two directions: the first one is to prove that the same holds for any Lindelöf $\Sigma$-space $X$ and hence $X^{\omega }$ is $d$-separable. We give an example of a countably compact space $X$ such that $X^{\omega }$ is not $d$-separable. On the other hand, we show that for any Lindelöf $p$-space $X$ there exists a discrete subset $D\subset X\times X$ such that $\Delta =\{(x,x):x\in X\}\subset \overline{D}$; in particular, the diagonal $\Delta$ is a retract of $\overline{D}$ and the projection...

It was proved in [HM] that each topological group (G,·,τ) may be embedded into a connected topological group (Ĝ,•,τ̂). In fact, two methods of introducing τ̂ were given. In this note we show relations between them.