A note on tameness of families having bounded variation

TL;DR

This paper investigates the properties of families of functions with bounded variation on linearly ordered sets, establishing their non-existence of independent sequences and proving sequential compactness results in various function spaces.

Contribution

It generalizes Helly's theorems to broader classes of functions and spaces, demonstrating sequential compactness for bounded variation and order-preserving functions.

Findings

Families with bounded variation lack independent sequences.

Function spaces of bounded variation are sequentially compact.

Order-preserving maps form sequentially compact spaces.

Abstract

We show that for arbitrary linearly ordered set $X$ any bounded family of (not necessarily, continuous) real valued functions on $X$ with bounded total variation does not contain independent sequences. We obtain generalized Helly's sequential compactness type theorems. One of the theorems asserts that for every compact metric space $(Y,d)$ the compact space $BV_r(X,Y)$ of all functions $X \to Y$ with variation $\leq r$ is sequentially compact in the pointwise topology. Another Helly type theorem shows that the compact space $M_+(X,Y)$ of all order preserving maps $X \to Y$ is sequentially compact where $Y$ is a compact metrizable partially ordered space in the sense of Nachbin.