Nets and Reverse Mathematics

Abstract

Nets are generalisations of sequences involving possibly uncountable index sets; this notion was introduced about a century ago by Moore and Smith, together with the generalisation to nets of various basic theorems of analysis due to Bolzano-Weierstrass, Dini, Arzela, and others. This paper deals with the Reverse Mathematics study of theorems about nets indexed by subsets of Baire space, i.e. part of third-order arithmetic. Perhaps surprisingly, over Kohlenbach’s base theory of higher-order Reverse Mathematics, the Bolzano-Weierstrass theorem for nets and the unit interval implies the Heine-Borel theorem for uncountable covers. Hence, the former theorem is extremely hard to prove (in terms of the usual hierarchy of comprehension axioms), but also unifies the concepts of sequential and open-cover compactness. Similarly, Dini’s theorem for nets is equivalent to the uncountable Heine-Borel theorem.