ИСТИНА

Let $X$ and $Y$ be topological spaces, and let $Z\subset X\times Y$. We say that {\it $Z$ is dense in $X\times Y$ along $Y$} if $Z\cap \{x\}\times Y$ is dense in $\{x\}\times Y$ for each $x\in X$. {\bf Theorem}. Suppose that $X$ is a strongly collectionwise normal space and the Hewitt extension $\mu X$ is a Lindel\"of $\Sigma$-space (e.g., $X$ is countably compact or $\sigma$-compact). Suppose also that $Y$ is a separable metrizable space and $Z$ is dense in $X\times Y$ along $Y$. Then the cellularity of the free topological group $F(Z)$ of $Z$ is countable. The special case of this theorem where $X$ is the one-point compactification of a discrete space was proved in [1]. Using methods of [1], we obtain the following example for $X=\omega_1$ and $Y$ is the space $\mathbb{P}$ of irrational numbers. {\bf Example.} Let $Z$ be a subspace of $\omega_1\times \mathbb{P}$ dense in $\omega_1\times \mathbb{P}$ along $\mathbb{P}$. Suppose that $|Z\cap \omega_1\times \{p\}|\le 1$ for all $p\in \mathbb{P}$. Then (1) $Z$ is a Mal'tsev space; (2) $Z$ is a locally countable locally metrizable submetrizable space and $c(Z)>\omega$; (3) $c(F(Z))\le\omega$; (4) $Z$ is not a retract of $F(Z)$ and, therefore, of any other topological group. Clearly, (4) follows from (2) and (3). [1] P.M Gartside, E.A. Reznichenko; O.V. Sipacheva, {\it Mal'tsev and retral spaces}, Topology Appl. \textbf{80} (1997), no. 1-2, 115--129.