About two Combinatorial Theorems

Abstract

We present two important theorems in combinatorial algebraic topology and convex combinatorial geometry, these are the nerve theorem and Helly’s theorem, giving examples of their use and relevance. We show that absolute extenders are equivalent to absolute retractions and that they are topological properties which allows, for example, to obtain triangulations for topological spaces expressed in terms of the rib of the associated simplicial complex. Thus also the abstract convex structures have main relevance for metrizable spaces, in particular the convex sets are absolute extensors and therefore retracted, thus being able to obtain regular coverings and good coverings. The intersection pattern of these coverings by convex gives rise to three important combinatorial numbers, the Helly number, Radon and Caratheodory. We conclude by making evident some combinatorial properties that these numbers possess, in particular that among the various uses of the Helly number.