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Displaying similar documents to “Pointwise convergence fails to be strict”

Rings of maps: sequential convergence and completion

Roman Frič (1999)

Czechoslovak Mathematical Journal

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The ring B ( R ) of all real-valued measurable functions, carrying the pointwise convergence, is a sequential ring completion of the subring C ( R ) of all continuous functions and, similarly, the ring 𝔹 of all Borel measurable subsets of R is a sequential ring completion of the subring 𝔹 0 of all finite unions of half-open intervals; the two completions are not categorical. We study ℒ 0 * -rings of maps and develop a completion theory covering the two examples. In particular, the σ -fields of sets form...

A completion of ℤ is a field

José E. Marcos (2003)

Czechoslovak Mathematical Journal

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We define various ring sequential convergences on ℤ and ℚ . We describe their properties and properties of their convergence completions. In particular, we define a convergence 𝕃 1 on ℤ by means of a nonprincipal ultrafilter on the positive prime numbers such that the underlying set of the completion is the ultraproduct of the prime finite fields ℤ / ( p ) . Further, we show that ( ℤ , 𝕃 1 * ) is sequentially precompact but fails to be strongly sequentially precompact; this solves a problem posed by D. Dikranjan. ...

Relatively coarse sequential convergence

Roman Frič, Fabio Zanolin (1997)

Czechoslovak Mathematical Journal

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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...