Bounded sets in spaces and topological groups (original) (raw)

Continuous Selections and Locally Pseudocompact Groups

Set-Valued Analysis, 2000

We characterize locally pseudocompact groups by means of the selection theory. Our result is the selection version of the well-known Comfort-Ross theorem on pseudocompactness which states that a topological group is pseudocompact if and only its Stone-Cech compactification is a topological group.

CLP-compactness for topological spaces and groups

Topology and its Applications, 2007

We study CLP-compact spaces (every cover consisting of clopen sets has a finite subcover) and CLP-compact topological groups. In particular, we extend a theorem on CLP-compactness of products from [J. Steprāns, A. Šostak, Restricted compactness properties and their preservation under products, Topology Appl. 101 (3) (2000) 213-229] and we offer various criteria for CLP-compactness for spaces and topological groups, that work particularly well for precompact groups. This allows us to show that arbitrary products of CLP-compact pseudocompact groups are CLP-compact. For every natural n we construct: (i) a totally disconnected, n-dimensional, pseudocompact CLP-compact group; and (ii) a hereditarily disconnected, n-dimensional, totally minimal, CLP-compact group that can be chosen to be either separable metrizable or pseudocompact (a Hausdorff group G is totally minimal when all continuous surjective homomorphisms G → H , with a Hausdorff group H , are open).

Bounded sets in topological groups and embeddings

Topology and its Applications, 2007

We show that the existence of a non-metrizable compact subspace of a topological group G often implies that G contains an uncountable supersequence (a copy of the one-point compactification of an uncountable discrete space). The existence of uncountable supersequences in a topological group has a strong impact on bounded subsets of the group. For example, if a topological group G contains an uncountable supersequence and K is a closed bounded subset of G which does not contain uncountable supersequences, then any subset A of K is bounded in G \ (K \ A). We also show that every precompact Abelian topological group H can be embedded as a closed subgroup into a precompact Abelian topological group G such that H is bounded in G and all bounded subsets of the quotient group G/H are finite. This complements Ursul's result on closed embeddings of precompact groups to pseudocompact groups. (M. Bruguera), mich@xanum.uam.mx (M. Tkachenko).

The three space problem in topological groups

Topology and its Applications, 2006

We study compact, countably compact, pseudocompact, and functionally bounded sets in extensions of topological groups. A property P is said to be a three space property if, for every topological group G and a closed invariant subgroup N of G, the fact that both groups N and G/N have P implies that G also has P. It is shown that if all compact (countably compact) subsets of the groups N and G/N are metrizable, then G has the same property. However, the result cannot be extended to pseudocompact subsets, a counterexample exists under p = c. Another example shows that extensions of groups do not preserve the classes of realcompact, Dieudonné complete and µ-spaces: one can find a pseudocompact, non-compact Abelian topological group G and an infinite, closed, realcompact 2000 Mathematics Subject Classification: Primary 54H11, 22A05; Secondary 54A20, 54G20.

On some kinds of factorizable topological groups

Cornell University - arXiv, 2022

Based on the concepts of R-factorizable topological groups and M-factorizable topological groups, we introduce four classes of factorizabilities on topological groups, named P M-factorizabilities, P m-factorizabilities, SM-factorizabilities and P SM-factorizabilities, respectively. Some properties of the four classes of spaces are investigated.

On zero-dimensionality and the connected component of locally pseudocompact groups

Proceedings of the American Mathematical Society

A topological group is locally pseudocompact if it contains a non-empty open set with pseudocompact closure. In this note, we prove that if G is a group with the property that every closed subgroup of G is locally pseudocompact, then G_0 is dense in the component of the completion of G, and G/G_0 is zero-dimensional. We also provide examples of hereditarily disconnected pseudocompact groups with strong minimality properties of arbitrarily large dimension, and thus show that G/G_0 may fail to be zero-dimensional even for totally minimal pseudocompact groups.

Weakly metrizable pseudocompact groups

Applied General Topology, 2006

We study various weaker versions of metrizability for pseudocompact abelian groups G: singularity (G possesses a compact metrizable subgroup of the form mG, m > 0), almost connectedness (G is metrizable modulo the connected component) and various versions of extremality in the sense of Comfort and co-authors (s-extremal, if G has no proper dense pseudocompact subgroups, r-extremal, if G admits no proper pseudocompact refinement). We introduce also weakly extremal pseudocompact groups (weakening simultaneously s-extremal and r-extremal). It turns out that this "symmetric" version of extremality has nice properties that restore the symmetry, to a certain extent, in the theory of extremal pseudocompact groups obtaining simpler uniform proofs of most of the known results. We characterize doubly extremal pseudocompact groups within the class of s-extremal pseudocompact groups. We give also a criterion for r-extremality for connected pseudocompact groups.

Abelian groups admitting a Fréchet–Urysohn pseudocompact topological group topology

Journal of Pure and Applied Algebra, 2010

We show that every Abelian group G with r 0 (G) = |G| = |G| ω admits a pseudocompact Hausdorff topological group topology T such that the space (G, T) is Fréchet-Urysohn. We also show that a bounded torsion Abelian group G of exponent n admits a pseudocompact Hausdorff topological group topology making G a Fréchet-Urysohn space if for every prime divisor p of n and every integer k ≥ 0, the Ulm-Kaplansky invariant f p,k of G satisfies (f p,k) ω = f p,k provided that f p,k is infinite and f p,k > f p,i for each i > k. Our approach is based on an appropriate dense embedding of a group G into a Σproduct of circle groups or finite cyclic groups.