Generalized Inverse Limits Indexed by Totally Ordered Sets

TL;DR

This paper explores generalized inverse limits indexed by totally ordered sets, extending classical theorems on connectedness and analyzing special cases with idempotent surjective set-valued functions.

Contribution

It generalizes inverse limit theory to totally ordered index sets and broadens connectedness results, including new theorems and examples for specific bonding functions.

Findings

Generalized connectedness theorems for inverse limits with totally ordered index sets

Extension of classical inverse limit theorems to broader index sets

Analysis of inverse limits with idempotent surjective set-valued functions

Abstract

Although inverse limits with factor spaces indexed by the positive integers are most commonly studied, Ingram and Mahavier have defined inverse limits with set-valued functions broadly enough for any directed index set to be used. In this paper, we investigate generalized inverse limits whose factor spaces are indexed by totally ordered sets. Using information about the projections of such inverse limits onto finitely many coordinates, we generalize various well-known theorems on connectedness in inverse limits. Moreover, numerous theorems and examples are given addressing the special case of an inverse limit with a single idempotent surjective u.s.c. bonding function.