Characterization of the Dedekind and countably Dedekind extensions of the lattice linear space of continuous bounded functions by means of order boundaries

In 1872 R. Dedekind constructed the set of real numbers $\mathbb{R}$ as a certain extension of the set of rational numbers $\mathbb{Q}$ by taking countable order regular cuts. This method was generalized and applied by G. MacNeille to some ordered mathematical systems. In this article the Dedekind – MacNeille method is applied to the mathematical system $C$ generated by the family $C_b(T,\mathcal{G})$ of all continuous bounded functions $f\colon T\to\mathbb{R}$ on the Tikhonov topological space $(T,\mathcal{G})$.
We consider Dedekind extension $C\!\longrightarrow D(C)$, and also countably Dedekind extension $C\!\longrightarrow D^0(C)$ as a closer analogue of the classical extension $\mathbb{Q}\!\longrightarrow\mathbb{R}$. Functional-factor descriptions of these extensions are given through families of functions uniform with respect to ensembles of subsets of the set $T$ having the Stone property and the Stone cozero property.
Characterizations of these extensions are given as some completions of the lattice linear space $C$ endowed with some local structure of ideal refinement.
The functional description and characterization of the countable Dedekind extension $C\!\longrightarrow D^0(C)$ turn out to be surprisingly similar with the functional description and characterization of the Riemannian extension $C\!\longrightarrow R_{\mu}$ generated by the factor-family of all functions on the Tikhonov space $(T,\mathcal{G})$ $\mu$-Riemann integrable with respect to a positive bounded Radon measure $\mu$.