Rationals with exotic convergences. II.
Relatively coarse sequential convergence
We generalize the notion of a coarse sequential convergence compatible with an algebraic structure to a coarse one in a given class of convergences. In particular, we investigate coarseness in the class of all compatible convergences (with unique limits) the restriction of which to a given subset is fixed. We characterize such convergences and study relative coarseness in connection with extensions and completions of groups and rings. E.g., we show that: (i) each relatively coarse dense group precompletion...
Relatively complete ordered fields without integer parts
We prove a convenient equivalent criterion for monotone completeness of ordered fields of generalized power series with exponents in a totally ordered Abelian group G and coefficients in an ordered field F. This enables us to provide examples of such fields (monotone complete or otherwise) with or without integer parts, i.e. discrete subrings approximating each element within 1. We include a new and more straightforward proof that is always Scott complete. In contrast, the Puiseux series field...
Rings of maps: sequential convergence and completion
The ring of all real-valued measurable functions, carrying the pointwise convergence, is a sequential ring completion of the subring of all continuous functions and, similarly, the ring of all Borel measurable subsets of is a sequential ring completion of the subring of all finite unions of half-open intervals; the two completions are not categorical. We study -rings of maps and develop a completion theory covering the two examples. In particular, the -fields of sets form an epireflective...
Size functions
We introduce the notion of a nonarchimedean size function similar to the notion of a size function introduced by Marcos. We describe a class of ring topologies on fields that are complete, neither first countable nor locally bounded, but have topologically nilpotent elements.
Some properties of linear right ideal nearrings.
Spectra of rings and lattices.
The lattice of precompact topologies on .
The open-open topology for function spaces.
Topological groups of divisibility
Topology on ordered fields
An ordered field is a field which has a linear order and the order topology by this order. For a subfield of an ordered field, we give characterizations for to be Dedekind-complete or Archimedean in terms of the order topology and the subspace topology on .
Totally bounded endomorphisms on a topological ring
Вложения в топологические поля и построение поля, пространство которого не нормально
О полноте свободного топологического модуля, порожденного дискретным пространством
О полноте топологического кольца многочленов
О продолжении кольцевой топологии -ограниченного поля на его простое трансцендентное расширение
О существовании свободных топологических модулей в классе модулей, являющихся ограниченными
О топологических свойствах гомоморфизмов универсальных алгебр
Продолжение топологии коммутативного кольца на некоторые его кольца частных