### On topologies of free groups

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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 $in\left({\left(H\right)}_{\tau}\right)\le {2}^{\tau \xb7in\left(H\right)}$ and $c\left({\left(H\right)}_{\tau}\right)\le {2}^{{2}^{\tau \xb7in\left(H\right)}}$ 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...

We consider $M$-mappings which include continuous mappings of spaces onto topological groups and continuous mappings of topological groups elsewhere. It is proved that if a space $X$ is an image of a product of Lindelöf $\Sigma $-spaces under an $M$-mapping then every regular uncountable cardinal is a weak precaliber for $X$, and hence $X$ has the Souslin property. An image $X$ of a Lindelöf space under an $M$-mapping satisfies $ce{l}_{\omega}X\le {2}^{\omega}$. Every $M$-mapping takes a $\Sigma \left({\aleph}_{0}\right)$-space to an ${\aleph}_{0}$-cellular space. In each of these results, the cellularity...

For every topological property $\mathcal{P}$, we define the class of $\mathcal{P}$-approximable spaces which consists of spaces X having a countable closed cover $\gamma $ such that the “section” $X(x,\gamma )=\bigcap \{F\in \gamma :x\in F\}$ has the property $\mathcal{P}$ for each $x\in X$. It is shown that every $\mathcal{P}$-approximable compact space has $\mathcal{P}$, if $\mathcal{P}$ is one of the following properties: countable tightness, ${\aleph}_{0}$-scatteredness with respect to character, $C$-closedness, sequentiality (the last holds under MA or ${2}^{{\aleph}_{0}}<{2}^{{\aleph}_{1}}$). Metrizable-approximable spaces are studied: every compact space in this class has...

It is well known that every $\mathbb{R}$-factorizable group is $\omega $-narrow, but not vice versa. One of the main problems regarding $\mathbb{R}$-factorizable groups is whether this class of groups is closed under taking continuous homomorphic images or, alternatively, whether every $\omega $-narrow group is a continuous homomorphic image of an $\mathbb{R}$-factorizable group. Here we show that the second hypothesis is definitely false. This result follows from the theorem stating that if a continuous homomorphic image of an $\mathbb{R}$-factorizable...

We present an example of a complete ${\aleph}_{0}$-bounded topological group $H$ which is not $\mathbb{R}$-factorizable. In addition, every ${G}_{\delta}$-set in the group $H$ is open, but $H$ is not Lindelöf.

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

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 $G$.

We introduce and study, following Z. Frol’ık, the class $\mathcal{B}\left(\mathcal{P}\right)$ 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 $\mathcal{B}\left(\mathcal{P}\right)$ is in $\mathcal{B}\left(\mathcal{P}\right)$ and that every regular Lindelöf $P$-space belongs to $\mathcal{B}\left(\mathcal{P}\right)$. It is also proved that all pseudo-${\aleph}_{1}$-compact $P$-groups are in $\mathcal{B}\left(\mathcal{P}\right)$. The problem of characterization of subgroups of $\mathbb{R}$-factorizable (equivalently, pseudo-${\aleph}_{1}$-compact) $P$-groups is considered as well. We give some necessary...

We show that subgroup of an $\mathbb{R}$-factorizable abelian $P$-group is topologically isomorphic to a subgroup of another $\mathbb{R}$-factorizable abelian $P$-group. This implies that closed subgroups of $\mathbb{R}$-factorizable $P$-groups are not necessarily $\mathbb{R}$-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 $\mathbb{R}$-factorizable $P$-groups are $\mathbb{R}$-factorizable and $\omega $-stable.

We study when a topological space has a weaker connected topology. Various sufficient and necessary conditions are given for a space to have a weaker Hausdorff or regular connected topology. It is proved that the property of a space of having a weaker Tychonoff topology is preserved by any of the free topological group functors. Examples are given for non-preservation of this property by “nice” continuous mappings. The requirement that a space have a weaker Tychonoff connected topology is rather...

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...

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