About two Combinatorial Theorems
DOI:
https://doi.org/10.15381/pesquimat.v24i1.19717Keywords:
convex structure, nerve, absolute retraction, d-representabilityAbstract
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.
Downloads
Published
Issue
Section
License
Copyright (c) 2021 Moisés Samuel Toledo Julián, Alex Molina Sotomayor, Napoleón Caro Tuesta
![Creative Commons License](http://i.creativecommons.org/l/by-nc-sa/4.0/88x31.png)
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
THE AUTHORS RETAIN THEIR RIGHTS:
a) The authors retain their trademark and patent rights, and also on any process or procedure described in the article.
b) The authors retain the right to share, copy, distribute, execute and publicly communicate the article published in Pesquimat magazine (for example, place it in an institutional repository or publish it in a book), with recognition of its initial publication in the Pesquimat magazine.
c) The authors retain the right to make a later publication of their work, to use the article or any part of it (for example: a compilation of their works, notes for conferences, thesis, or for a book), provided that they indicate the source of publication (authors of the work, magazine, volume, number and date).