Characterization of the Dedekind and countably Dedekind extensions of the lattice linear space of continuous bounded functions by means of order boundaries
Abstract:
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 $\mu$.
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