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The Bohr compactification, modulo a metrizable subgroup

W. Comfort, F. Trigos-Arrieta, S. Wu (1993)

Fundamenta Mathematicae

The authors prove the following result, which generalizes a well-known theorem of I. Glicksberg [G]: If G is a locally compact Abelian group with Bohr compactification bG, and if N is a closed metrizable subgroup of bG, then every A ⊆ G satisfies: A·(N ∩ G) is compact in G if and only if {aN:a ∈ A} is compact in bG/N. Examples are given to show: (a) the asserted equivalence can fail in the absence of the metrizability hypothesis, even when N ∩ G = {1}; (b) the asserted equivalence can hold for suitable...

The controlled separable projection property for Banach spaces

Jesús Ferrer, Marek Wójtowicz (2011)

Open Mathematics

Let X, Y be two Banach spaces. We say that Y is a quasi-quotient of X if there is a continuous operator R: X → Y such that its range, R(X), is dense in Y. Let X be a nonseparable Banach space, and let U, W be closed subspaces of X and Y, respectively. We prove that if X has the Controlled Separable Projection Property (CSPP) (this is a weaker notion than the WCG property) and Y is a quasi-quotient of X, then the structure of Y resembles the structure of a separable Banach space: (a) Y/W is norm-separable...

The covering property for σ-ideals of compact, sets

Carlos Uzcátegui (1992)

Fundamenta Mathematicae

The covering property for σ-ideals of compact sets is an abstract version of the classical perfect set theorem for analytic sets. We will study its consequences using as a paradigm the σ-ideal of countable closed subsets of 2 ω .

The dual group of a dense subgroup

William Wistar Comfort, S. U. Raczkowski, F. Javier Trigos-Arrieta (2004)

Czechoslovak Mathematical Journal

Throughout this abstract, G is a topological Abelian group and G ^ is the space of continuous homomorphisms from G into the circle group 𝕋 in the compact-open topology. A dense subgroup D of G is said to determine G if the (necessarily continuous) surjective isomorphism G ^ ↠ D ^ given by h ↦ h | D is a homeomorphism, and G is determined if each dense subgroup of G determines G . The principal result in this area, obtained independently by L. Außenhofer and M. J. Chasco, is the following: Every metrizable group is...

The existence of initially ω 1 -compact group topologies on free Abelian groups is independent of ZFC

Artur Hideyuki Tomita (1998)

Commentationes Mathematicae Universitatis Carolinae

It was known that free Abelian groups do not admit a Hausdorff compact group topology. Tkachenko showed in 1990 that, under CH, a free Abelian group of size ℭ admits a Hausdorff countably compact group topology. We show that no Hausdorff group topology on a free Abelian group makes its ω -th power countably compact. In particular, a free Abelian group does not admit a Hausdorff p -compact nor a sequentially compact group topology. Under CH, we show that a free Abelian group does not admit a Hausdorff...

The Gruenhage property, property *, fragmentability, and σ-isolated networks in generalized ordered spaces

Harold Bennett, David Lutzer (2013)

Fundamenta Mathematicae

We examine the Gruenhage property, property * (introduced by Orihuela, Smith, and Troyanski), fragmentability, and the existence of σ-isolated networks in the context of linearly ordered topological spaces (LOTS), generalized ordered spaces (GO-spaces), and monotonically normal spaces. We show that any monotonically normal space with property * or with a σ-isolated network must be hereditarily paracompact, so that property * and the Gruenhage property are equivalent in monotonically normal spaces....

Tightness and π-character in centered spaces

Murray Bell (1999)

Colloquium Mathematicae

We continue an investigation into centered spaces, a generalization of dyadic spaces. The presence of large Cantor cubes in centered spaces is deduced from tightness considerations. It follows that for centered spaces X, πχ(X) = t(X), and if X has uncountable tightness, then t(X) = supκ : 2 κ ⊂ X. The relationships between 9 popular cardinal functions for the class of centered spaces are justified. An example is constructed which shows, unlike the dyadic and polyadic properties, that the centered...

Tightness of compact spaces is preserved by the t -equivalence relation

Oleg Okunev (2002)

Commentationes Mathematicae Universitatis Carolinae

We prove that if there is an open mapping from a subspace of C p ( X ) onto C p ( Y ) , then Y is a countable union of images of closed subspaces of finite powers of X under finite-valued upper semicontinuous mappings. This allows, in particular, to prove that if X and Y are t -equivalent compact spaces, then X and Y have the same tightness, and that, assuming 2 𝔱 > 𝔠 , if X and Y are t -equivalent compact spaces and X is sequential, then Y is sequential.

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