Displaying similar documents to “A completion of ℤ is a field”

Pointwise convergence fails to be strict

Ján Borsík, Roman Frič (1998)

Czechoslovak Mathematical Journal

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It is known that the ring B ( ℝ ) of all Baire functions carrying the pointwise convergence yields a sequential completion of the ring C ( ℝ ) of all continuous functions. We investigate various sequential convergences related to the pointwise convergence and the process of completion of C ( ℝ ) . In particular, we prove that the pointwise convergence fails to be strict and prove the existence of the categorical ring completion of C ( ℝ ) which differs from B ( ℝ ) .

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...

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...