Nets and Reverse Mathematics

摘要: 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 various basic theorems analysis due Bolzano-Weierstrass, Dini, Arzela, others. This paper deals Reverse Mathematics study indexed subsets Baire space, i.e. part third-order arithmetic. Perhaps surprisingly, over Kohlenbach’s base theory higher-order Mathematics, Bolzano-Weierstrass theorem for unit interval implies Heine-Borel covers. Hence, former is extremely hard prove (in terms usual hierarchy comprehension axioms), but also unifies concepts sequential open-cover compactness. Similarly, Dini’s equivalent theorem.