FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

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

**Topological Helly theorem**

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 $$**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 $k$£ 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 $\{k$-2,¼,
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 $k$£ 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 *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.

Location: http://mech.math.msu.su/~fpm/eng/k02/k022/k02204h.htm

Last modified: November 26, 2002