  Nets are generalisations of sequences involving possibly uncountable index
sets; this notion was introduced about a century ago by Moore and Smith. They
also established the generalisation to nets of various basic theorems of
analysis due to Bolzano-Weierstrass, Dini, Arzela, and others. More recently,
nets are central to the development of domain theory, providing intuitive
definitions of the associated Scott and Lawson topologies, among others. This
paper deals with the Reverse Mathematics study of basic theorems about nets. We
restrict ourselves to nets indexed by subsets of Baire space, and therefore
third-order arithmetic, as such nets suffice to obtain our main results. Over
Kohlenbach's base theory of higher-order Reverse Mathematics, the
Bolzano-Weierstrass theorem for nets implies the Heine-Borel theorem for
uncountable covers. We establish similar results for other basic theorems about
nets and even some equivalences, e.g. for Dini's theorem for nets. Finally, we
show that replacing nets by sequences is hard, but that replacing sequences by
nets can obviate the need for the Axiom of Choice, a foundational concern in
domain theory. In an appendix, we study the power of more general index sets,
establishing that the 'size' of a net is directly proportional to the power of
the associated convergence theorem.
