I PRIKLADNAYA MATEMATIKA
(FUNDAMENTAL AND APPLIED MATHEMATICS)
2002, VOLUME 8, NUMBER 2, PAGES 365-405
S. A. Bogatyi
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We give an axiomatic version of topological Helly theorem, from which we derive many corollaries about common intersection (union).
Instead of the space
For any closed covering of
Our results are valid for arbitrary normal space of finite cohomological dimension, but are partially new even in the case of the plane. In particular, we fill the gap in the topological Helly theorem of 1930 for plane singular cells. If in the family of plane compacta the union of any 2 compacta is path-connected, and the union of any 3 compacta is simply connected, then the total intersection of all compacta of the family is non-empty. It is shown that if in the family of plane simply connected Peano continua the intersection of any 2 continua is connected and the intersection of any 3 continua is non-empty, then any compactum obtained from the compacta of the family with the operations of union and intersection is a non-empty simply connected Peano continuum. Analogously, if in the family of plane simply connected Peano continua the union of any 2 and any 3 continua is a simply connected Peano continuum, then any compactum obtained from the compacta of the family with the operations of union and intersection is a non-empty simply connected Peano continuum. Analogous statements are true for continua that do not separate the plane.
All articles are published in Russian.
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Last modified: November 26, 2002