ИСТИНА

A topological space $X$ is said to be {\it non-Archimedean} if $X$ has a {\it non-Archimedean} base $\cal B$; this means that if $U,V\in\cal B$ and $U\cap V\ne\emptyset$, then either $U\subset V$ or $V\subset U$. Free topological groups of non-Archimedean spaces were studied in [1], where it was proved that any non-Archimedean space $X$ is a retract of its free topological group $F(X)$ (therefore, $c(F(X))\ge c(X)$). {\bf Theorem} For any non-Archimedean topological space $X$, $c(F(X)) = w(X)$. Given a tree $T$, let $X_T$ denote the branch set of $T$. The sets of the form $U_t=\{x\in X_T:t\in x\}$ form a base of a non-Archimedean topology on $X_T$; we assume $X_T$ to be endowed with this topology. If there exists a Souslin tree $T$, then $c(X_T)\le\omega$ and, by the above theorem, $c(F(X_T))>\omega$. Note that Martin's axiom and the negation of the continuum hypothesis imply that if $c(X)\le\omega$, then $c(F(X))\le\omega$. [1] P. M. Gartside, E. A. Reznichenko, and O. V. Sipacheva, {\it Mal'tsev and retral spaces}, Topology Appl. \textbf{80} (1997), no. 1-2, 115--129.