A classification of the cofinal structures of precompacta

TL;DR

This paper classifies the cofinal structures of precompact and compact sets in metric spaces and applies this to understand local bases in groups of continuous functions.

Contribution

It provides a complete classification of cofinal structures of precompact and compact sets and applies this to the structure of local bases in function groups.

Findings

Classified cofinal structures of precompact and compact sets in metric spaces

Applied classification to local bases in groups of continuous functions

Enhanced understanding of the topology of function spaces

Abstract

We provide a complete classification of the possible cofinal structures of the families of precompact (totally bounded) sets in general metric spaces, and compact sets in general complete metric spaces. Using this classification, we classify the cofinal structure of local bases in the groups $\C(X,\bbR)$ of continuous real-valued functions on complete metric spaces $X$, with respect to the compact-open topology.