ИСТИНА

Let us say that a set $X$ in a group $G$ is \emph{$k$-independent} if $x_1^{\epsilon_1} x_2^{\epsilon_2} \dots x_k^{\epsilon_k}\ne 1$ for any $\epsilon_i=\pm1$ and any different $x_i\in X$, and that $X$ is \emph{independent} if it is $k$-independent for any $k\ge 2$. Independent sets and their generalizations naturally arise in studying topological groups with extremal properties on the one hand and large sets in groups on the other. Of special interest is the existence of nonclosed or nonclosed discrete independent sets. For example, we prove that \begin{enumerate} \item[(i)] the existence of a countable extremally disconnected group containing a nonclosed 4-independent set implies that of rapid (ultra)filters; \item[(ii)] the existence of a Ramsey ultrafilter on a set of cardinality $\kappa$ implies that of a Boolean topological group of dispersion character $\kappa$ in which all independent sets are closed (and discrete); \item[(iii)] if a Boolean topological group contains a nonclosed independent set and zero is not a limit point for any 3-independent set, then there exists a 3-arrow ultrafilter; \item[(iv)] each Boolean topological group contains a closed discrete maximal independent set. \end{enumerate} The proofs heavily depend on Ramsey theory. We also discuss generalizations of independent sets and a relationship between the presence of nonclosed generalized independent sets in topological groups, topological properties of large sets in these groups, and the existence of Ramsey-type ultrafilters. Independent sets are also very useful in studying the cardinal functions $\delta$ and $\hat\delta$ on topological groups. Given a topological space $X$ and a point $x\in X$, we set $$ \begin{aligned} \delta(x, X) = \sup\{\kappa: &\text{ there exist $\kappa$ disjoint discrete sets}\\ &\text{ for each of which $x$ is a unique limit point}\},\\ \hat\delta(x, X) = \sup\{\kappa: &\text{ there exist closed sets $C_\alpha$, $\alpha<\kappa$, such that}\\ &\text{ $x$ is a limit point of each $C_\alpha$ and }\\ &\text{ $C_\alpha\cap C_\beta =\{x\}$ for any $\alpha, \beta<\kappa$, $\alpha\ne \beta$}\}, \end{aligned} $$ and $$ \delta(X)=\sup\{\delta(x, X): x\in X\},\qquad \hat\delta(X)=\sup\{\hat\delta(x, X): x\in X\}. $$ Clearly, $1\le\delta(x,X)\le\hat\delta(x,X)$ for any space $X$ and any $x\in X$; we also have $\hat\delta(x, X)=1$ for any $F$-space (and hence any extremally disconnected space) $X$ and any $G_\delta$-point $x\in X$. We show, in particular, that $\hat\delta(G)\ge 2$ for any topological group containing a closed independent set with more than two limit points (it follows that the existence of an extremally disconnected group $G$ with $\hat\delta(G)>1$ implies that of measurable cardinals) and that $\delta(G)>1$ for any countable topological group containing a nonclosed 4-independent set $A$, provided that $A$ has a limit point with nonrapid filter of neighborhoods. Some other related results are also discussed.