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We define various ring sequential convergences on and . We describe their properties and properties of their convergence completions. In particular, we define a convergence on 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 . Further, we show that is sequentially precompact but fails to be strongly sequentially precompact; this solves a problem posed by D. Dikranjan.
In this paper, it is proved that a first-countable paratopological group has a regular -diagonal, which gives an affirmative answer to Arhangel’skii and Burke’s question [Spaces with a regular -diagonal, Topology Appl. 153 (2006), 1917–1929]. If is a symmetrizable paratopological group, then is a developable space. We also discuss copies of and of in paratopological groups and generalize some Nyikos [Metrizability and the Fréchet-Urysohn property in topological groups, Proc. Amer. Math....
Let be a uniformly closed and locally m-convex -algebra. We obtain internal conditions on stated in terms of its closed ideals for to be isomorphic and homeomorphic to , the -algebra of all the real continuous functions on a normal topological space endowed with the compact convergence topology.
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 of idempotents in , 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....
We prove a general theorem about preservation of the covering dimension by certain covariant functors that implies, among others, the following concrete results.
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If
G
is a pathwise connected separable metric...
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