On completeness in a non-Archimedean setting via firm reflections
| dc.creator | Deses, D. | |
| dc.creator | Lowen-Colebunders, E. | |
| dc.date | 2004-02-17 | |
| dc.date.accessioned | 2026-07-07T05:05:31Z | |
| dc.date.available | 2026-07-07T05:05:31Z | |
| dc.description | We develop a completion theory for (general) non-Archimedean spaces based on the theory on "a categorical concept of completion of objects" as introduced by G.C.L. Brümmer and E. Giuli. Our context is the construct $\mathbf{NA}_0$ of all Hausdorff non-Archimedean spaces and uniformly continuous maps and $\mathcal{V}$ is the class of all epimorphic embeddings in $\mathbf{NA}_0$. We determine the class ${\bf Inj} \mathcal{V}$ of all $\mathcal{V}$-injective objects and we present an internal characterization as "complete objects". The basic tool for this characterization is a notion of small collections that in some sense preserve the inclusion order on the non-Archimedean structure. We prove that the full subconstruct $\mathbf{CNA}_0$ consisting of all complete objects forms a firmly $\mathcal{V}$-reflective subcategory. This means that every object $X$ in $\mathbf{NA}_0$ has a completion which is a $\mathcal{V}$-reflection $r_X:X\to RX$ into the full subconstruct $\mathbf{CNA}_0$ of "complete spaces". Moreover this completion is unique (up to isomorphism) in the sense that, considering $L(\mathbf{CNA}_0)$, the class of all those morphisms $u: X\to Y$ for which $Ru:RX\to RY$ is an isomorphism, one has that $\mathcal{V}$ is contained in $L(\mathbf{CNA}_0)$. In fact one even has $\mathcal{V}=L(\mathbf{CNA}_0)$. Finally we apply our constructions to the classical case of Hausdorff non-Archimedean uniform spaces, in that case our completion reduces to the standard one. | |
| dc.description | Special volume: p-adic numbers in number theory, analytic geometry and funtional analysis | |
| dc.identifier | https://arxiv.org/abs/math/0402275 | |
| dc.identifier | http://arxiv.org/abs/math/0402275 | |
| dc.identifier | Bulletin of the Belgian Mathematical Society (2002), 49--61 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/70193 | |
| dc.subject | General Topology | |
| dc.subject | Category Theory | |
| dc.subject | 54E15, 54B30, 54D35, 26E30, 18G05 | |
| dc.title | On completeness in a non-Archimedean setting via firm reflections | |
| dc.type | text |