Ultimately increasing functions

Abstract

A function gg between directed sets ⟨Σ,⪰′⟩⟨Σ,⪰′⟩ and ⟨Λ,⪰⟩⟨Λ,⪰⟩ is called \emph{ultimately increasing} if for each σ1∈Σσ1∈Σ there exists σ2⪰′σ1σ2⪰′σ1 such that σ⪰′σ2⇒g(σ)⪰g(σ1)σ⪰′σ2⇒g(σ)⪰g(σ1). A subnet of a net aa defined on ⟨Λ,⪰⟩⟨Λ,⪰⟩ \cite {Ke} is nothing but a composition of the form a∘ga∘g where gg is ultimately increasing and g(Σ)g(Σ) is a cofinal subset of ΛΛ. While even for linearly ordered sets, an increasing net defined on a cofinal subset of the domain need not have an increasing extension, in complete generality, it must have an ultimately increasing extension, and conversely when the domain is linearly ordered. Applications are given in the context of functions with values in a linearly ordered set equipped with the order topology - in particular, the extended real numbers. For example, we show that a real sequence ⟨an⟩⟨an⟩ converges to the supremum of its set of terms if and only if ⟨an⟩⟨an⟩ is the supremum of the ultimately increasing sequences that it majorizes.