FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2002, VOLUME 8, NUMBER 2, PAGES 365-405

S. A. Bogatyi

Abstract

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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 $\mathbb R^m$ we consider an arbitrary
normal space $X$ with cohomological dimension not greater than $m$
and with trivial $m$ -dimensional cohomological group.
Instead of the convex subsets we consider closed acyclic
subsets and instead of the conditions on intersections we
impose (obtain) conditions on the values of arbitrary simple
Boolean functions. In the extreme cases (only unions or
intersections are considered) the conditions have the following
form: for any $k$ sets of the given family, for $k \leq m+1$ ,
either their common intersection has trivial cohomologies in
all dimensions not greater than $m-k$ , or their common union has trivial
cohomologies in all dimensions from $\{k-2,\ldots,m-1\}$ . Then
it is proved that any subset obtained from sets of the given family
with operations of union and intersection is nonempty and
acyclic.

For any closed covering of $m$ -dimensional sphere the
intersection of some $m+2$ elements is empty or for some $k \leq m+1$
there exist $k$ elements of the covering such that their
intersection has non-trivial $(m+1-k)$ -dimensional
cohomologies.

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 \emph{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.

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Last modified: November 26, 2002