ИСТИНА

Let $\kappa$ be any infinite cardinal. Following Szyma\'nski, we use the notation $\Seq_\kappa$ for the set of all finite sequences of ordinals in $\kappa$. For any filter $\mathscr F$ on $\kappa$, Szyma\'nski defined a topology $T_{\mathcal F}$ on $\Seq_\kappa$, declaring $U\subset \Seq_\kappa$ to be open if and only if $$ \forall s\in \Seq_\kappa (s\in U\to \{\alpha\in \kappa: s\,{}^\frown \alpha\in U)\in \mathscr F). $$ He proved that if $\mathscr U$ is a nonprincipal ultrafilter on $\kappa$, then the space $\Seq_\kappa(\mathscr U)=(\Seq_\kappa, T_{\mathscr U})$ has no isolated points and is completely regular and extremally disconnected. Then, Kato showed that $\Seq_\omega (\mathscr U)$ is homogeneous for any nonprincipal ultrafilter on $\omega$, and Vaughan noted that the space $\Seq_\omega(\mathscr U)$ is homeomorphic to its subspace $\Seq^\uparrow_\omega(\mathscr U)$ consisting of increasing sequences and to the space $[\omega]^{<\omega}$ endowed with the topology in which a set $U$ is open if and only if $$ \forall F\in [\omega]^{<\omega} (F\in U\to \{n\in \omega: F\vartriangle\{n\}\in U)\in \mathscr U); $$ it is natural to refer to this topology and its natural extension to uncountable cardinals as the \emph{symmetric version} of $T_{\mathscr U}$. The topology $T_{\mathscr U}$ on $\Seq_\omega$ was systematically studied by Todorcevic, who called it the \emph{$\mathscr U$-topology}. Its definition (generalized naturally to arbitrary cardinals) is based on the notion of a \emph{$\mathscr U$-tree}, that is, a set $T\subset \Seq_\kappa$ closed under initial segments and such that $$ \text{for each $t\in T$,}\quad T(t)=\{n\in \kappa: t\,{}^\frown n\in T\}\in \mathscr U $$ (here ${}^\frown$ denotes concatenation). \begin{theorem*} \label{U-topology} For an uncountable cardinal $\kappa$ and a uniform ultrafilter $\mathscr U$ on $\kappa$, the following conditions are equivalent: \begin{enumerate} \item[(i)] $\bigoplus^\kappa \mathbb Z_2\cong \Seq^\uparrow_\kappa$ with the $\mathscr U$-topology is an extremally disconnected topological group with linear topology; \item[(ii)] the $\mathscr U$-topology on $\bigoplus^\kappa \mathbb Z_2\cong \Seq^\uparrow_\kappa$ is a group topology; \item[(iii)] $[\kappa]^{<\omega}$ with the symmetric $\mathscr U$-topology is an extremally disconnected topological group with linear topology; \item[(iv)] the symmetric $\mathscr U$-topology on $[\kappa]^{<\omega}$ is a group topology; \item[(v)] $\mathscr U$ is a Ramsey ultrafilter. \end{enumerate} \end{theorem*} It follows, in particular, that the existence of measurable cardinals implies that of nondiscrete extremally disconnected groups, but it is still unclear whether such groups exist in ZFC. Malykhin proved that the existence of a nondiscrete extremally disconnected group is equivalent to the existence of such a Boolean group. Moreover, all known (consistent) examples of extremally disconnected groups have linear topology. \begin{theorem*} Let $G$ be a Boolean topological group with zero element $\zero$ which is an $F'$-space and contains a countable family of $\{H_n:n\in\omega\}$ of open subgroups such that $\bigcap H_n=\{\zero\}$. Then $G$ has an open subgroup admitting a continuous isomorphism onto a subgroup of the countable Cartesian product of discrete countable Boolean groups with the product topology. \end{theorem*} \begin{corollary*} If there exists a nondiscrete extremally disconnected group of Ulam nonmeasurable cardinality with linear topology, then there exists a nondiscrete extremally disconnected Boolean group with linear topology admitting a continuous isomorphism to a subgroup of $\mathbb Z_2^\omega$ with the product topology. \end{corollary*} \begin{summary*} If there exists in ZFC a nondiscrete extremally disconnected group with linear topology, then there must exist such a group $G$ with all of the following properties: \begin{itemize} \item \emph{$G$ is Boolean}; \item \emph{$G$ is uncountable, and all countable subsets of $G$ are closed and discrete}; \item \emph{$G$ contains a countable family of open subgroups with trivial intersection}; \item $|G|\le 2^\omega$; \item \emph{Given any basis $E$ of $G$, there exists a $k\in \omega$ for which $\zero \in [E]^k$}; \item \emph{There exists a basis $E=\{e_\alpha : \alpha<|G|\}$ of $G$ such that all subgroups $H_\alpha=\langle \{e_\beta : \beta\ge\alpha\}\rangle $ are dense in $G$.} \end{itemize}