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A completion of ℤ is a field

José E. Marcos (2003)

Czechoslovak Mathematical Journal

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.

A note on paratopological groups

Chuan Liu (2006)

Commentationes Mathematicae Universitatis Carolinae

In this paper, it is proved that a first-countable paratopological group has a regular G δ -diagonal, which gives an affirmative answer to Arhangel’skii and Burke’s question [Spaces with a regular G δ -diagonal, Topology Appl. 153 (2006), 1917–1929]. If G is a symmetrizable paratopological group, then G is a developable space. We also discuss copies of S ω and of S 2 in paratopological groups and generalize some Nyikos [Metrizability and the Fréchet-Urysohn property in topological groups, Proc. Amer. Math....

Closed ideals in topological algebras: a characterization of the topological Φ -algebra C k ( X )

F. Montalvo, Antonio A. Pulgarín, Batildo Requejo Fernández (2006)

Czechoslovak Mathematical Journal

Let A be a uniformly closed and locally m-convex Φ -algebra. We obtain internal conditions on A stated in terms of its closed ideals for A to be isomorphic and homeomorphic to C k ( X ) , the Φ -algebra of all the real continuous functions on a normal topological space X endowed with the compact convergence topology.

Longer chains of idempotents in βG

Neil Hindman, Dona Strauss, Yevhen Zelenyuk (2013)

Fundamenta Mathematicae

Given idempotents e and f in a semigroup, e ≤ f if and only if e = fe = ef. We show that if G is a countable discrete group, p is a right cancelable element of G* = βG∖G, and λ is a countable ordinal, then there is a strictly decreasing chain ⟨ q σ ⟩ σ < λ of idempotents in C p , the smallest compact subsemigroup of G* with p as a member. We also show that if S is any infinite subsemigroup of a countable group, then any nonminimal idempotent in S* is the largest element of such a strictly decreasing chain of idempotents....

Module-valued functors preserving the covering dimension

Jan Spěvák (2015)

Commentationes Mathematicae Universitatis Carolinae

We prove a general theorem about preservation of the covering dimension dim by certain covariant functors that implies, among others, the following concrete results. If G G is a pathwise connected separable metric...

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