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Topological Helly theorem. (Russian. English summary) Zbl 1028.52004
An axiomatic version of the classical Helly theorem about the intersections of convex subsets in \({\mathbb R}^m\) is given. Various corollaries concerning geometrical and topological Helly theorems are deduced. It is shown that different forms of the Helly theorem are equivalent to each other and correspond to different representations of equivalent conditions. The author considers closed acyclic subspaces in an arbitrary normal space \(X\) of cohomological dimension less than or equal to \(m\) and with trivial \(m\)-dimensional cohomology group. The conditions are given in terms of arbitrary monotone Boolean functions (i.e., functions including only union and intersection operations). In the extreme cases (only unions or only intersections are considered) the conditions have the following form: for any \(k\) subsets of the given family, \(k\leq m+1\), either their common intersection has trivial cohomology in all dimensions not greater than \(m-k\), or their common union has trivial cohomology in all dimensions from \(k-2\) to \(m-1\). Then it is proved that any subset obtained from sets of the given family with operations of union and intersection is nonempty and acyclic.
Even in the case of the plane the author obtains some new results. For example, he partially fills the gap in the topological Helly theorem of 1930 for the plane singular case.
In particular some corollaries for plane compacta, Peano continuum, continua that do not separate the plane, unicoherent continua, etc. are obtained.

52A35 Helly-type theorems and geometric transversal theory
55M99 Classical topics in algebraic topology
54F15 Continua and generalizations