Chapter 7 - Nets and Convergences

Abstract

This chapter provides an overview on the convergence of sequences in metric spaces. Nets are a generalization of sequences. A sequence is a function whose domain is N; a net (or “generalized sequence”) is a function whose domain is any directed set. A convergence space is a set X equipped with some rule that specifies which nets or equivalently that filters—converge to which “limits” in X. Analysts who are already familiar with convergent sequences in metric spaces should have little difficulty with convergent nets; as is shown in this chapter, nets and convergence spaces are natural generalizations of sequences in metric spaces. Nets are particularly helpful in understanding topologies that are known to be nonmetrizable (such as the weak topology of an infinite-dimensional normed vector space), or those that are not known to be metrizable..