ИСТИНА

A topological space is said to be extremally disconnected if the closure of any open set in this space is open. So far, no example of a nondiscrete extremally disconnected topological group in ZFC has been constructed (this is Arhangelskii's problem [1]), although there exist consistent examples [2--6]. Malykhin noticed that any extremally disconnected group must contain an open Boolean subgroup [4]. As is known, all Boolean groups are algebraically free. Thus, if there exists an extremally disconnected group, then there exists such a group being the free Boolean group $B(X)$ generated by some space $X$ (with the induced topology $\mathcal T_X$). The topology of this group must not be free, i.e., it may not be the strongest group topology inducing $\mathcal T_X$ (although, most of the existing examples of extremally disconnected groups are the free Boolean groups with the free or the free linear topology over ultrafilters or their modifications; following Malykhin, we say that a group topology is linear if it has a neighborhood base at zero formed by subgroups). We prove, in particular, that a ZFC example of an extremally disconnected group topology cannot be neither free nor free linear; to be more precise, the nonexistence of a nondiscrete extremally disconnected free Boolean group with the free group or the free linear precompact group topology (with respect to some basis) or any topology intermediate between these two is consistent with ZFC. Any extremally disconnected group topology on a countable Boolean group generates a family of ultrafilters. For example, if $\{U_n: n\in \omega\}$ is a decreasing sequence of clopen neighborhoods of zero such that $\bigcap U_n=\{0\}$, then extremal disconnectedness implies that the sets $\{n\in \omega: (U_n\setminus U_{n+1})\cap U\ne \emptyset\}$, where $U$ ranges over all open neighborhoods of zero, form an ultrafilter. We describe some properties of the ultrafilters thus arising; in particular, we show that not every ultrafilter can be obtained in this way. Purely topological properties of extremally disconnected groups and groups related to them are also considered. Open problems are posed. REFERENCES [1] A. V. Arkhangel'skii, ``Extremally disconnected compact Hausdorff spaces of weight $\mathfrak c$ are inhomogeneous,'' Dokl. Akad. Nauk SSSR 75, 751--754 (1967). [2] S. Sirota, ``Products of topological groups and extremal disconnectedness,'' Mat. Sb. 79 (2), 179--192 (1969). [3] A. Louveau, ``Sur un article de S. Sirota,'' Bull. Sci. Math. (France) 96, 3--7 (1972). [4] V. I. Malykhin, ``Extremally disconnected and nearly extremally disconnected groups,'' Dokl. Akad. Nauk SSSR, 220, 27--30 (1975). [5] E. G. Zelenyuk, ``Topological groups with finite semigroups of ultrafilters,'' Mat. Studii 6, 41--52 (1996). [6] E. G. Zelenyuk, ``Extremal ultrafilters and topologies on groups,'' Mat. Studii 14, 121--140 (2000).