Rational Extensions of C(X) via Hausdorff Continuous Functions

TL;DR

This paper extends the structure of continuous functions on a space to a broader class of Hausdorff continuous interval functions, showing they form a complete ring and represent quotients and completions of the original function ring.

Contribution

It introduces nearly finite Hausdorff continuous functions as an extension of C(X), establishing their algebraic and topological completeness and their role in representing quotients and completions.

Findings

Houses the ring of quotients within Hausdorff continuous functions.

Proves the completeness of the extended function space.

Shows the metric and algebraic structures are preserved.

Abstract

The ring operations and the metric on $C(X)$ are extended to the set $\mathbb{H}_{nf}(X)$ of all nearly finite Hausdorff continuous interval valued functions and it is shown that $\mathbb{H}_{nf}(X)$ is both rationally and topologically complete. Hence, the rings of quotients of $C(X)$ as well as their metric completions are represented as rings of Hausdorff continuous functions.