Dense subsets of maximally almost periodic groups

A (discrete) group is said to be maximally almost periodic if the points of are distinguished by homomorphisms into compact Hausdorff groups. A Hausdorff topology on a group is totally bounded if whenever there is such that . For purposes of this abstract, a family with a totally bounded topological group is a strongly extraresolvable family if (a)  \vert G\vert$">, (b) each is dense in , and (c) distinct satisfy ; a totally bounded topological group with such a family is a strongly extraresolvable topological group.

We give two theorems, the second generalizing the first.

Theorem 1. Every infinite totally bounded group contains a dense strongly extraresolvable subgroup.

Corollary. In its largest totally bounded group topology, every infinite Abelian group is strongly extraresolvable.

Theorem 2. Let be maximally almost periodic. Then there are a subgroup of and a family such that

(i) is dense in every totally bounded group topology on ;

(ii) the family is a strongly extraresolvable family for every totally bounded group topology on such that ; and

(iii) admits a totally bounded group topology as in (ii).

Remark. In certain cases, for example when is Abelian, one must in Theorem 2 choose . In certain other cases, for example when the largest totally bounded group topology on is compact, the choice is impossible.

     

The degree of subsequentiality of a subsequential filter

A space is called subsequential if it is a subspace of a sequential space. A free filter F on ω is called subsequential if the space ω∪{F} is subsequential. The purpose of this paper is to introduce the degree of subsequentiality of a subsequential filter in a similar way as it is done in the realm of sequential spaces. A method to produce subsequential filters with arbitrary subsequential degree is given.     

Subsequential filters

Following N. Noble, we say that a space is subsequential if it is a subspace of a sequential space. A free filter F on ω is called subsequential if the space ω∪{F} is subsequential. In this paper, we state several properties of these filters.     

Erratum to ``Baire spaces and Vietoris hyperspaces'

The main purpose of this short note is to correct an error in ``Baire spaces and Vietoris hyperspaces', Proc. Amer. Math. Soc. 135 (2007), no. 1, 299-303.

     

Some remarks on the product of two C α-compact subsets

For a cardinal , we say that a subset B of a space X is C -compact in X if for every continuous function is a compact subset of . If B is a C-compact subset of a space X, then (B, X) denotes the degree of C -compactness of B in X. A space X is called -pseudocompact if X is C -compact into itself. For each cardinal , we give an example of an -pseudocompact space X such that X × X is not pseudocompact: this answers a question posed by T. Retta in Some cardinal generalizations of pseudocompactness Czechoslovak Math. J. 43 (1993), 385–390. The boundedness of the product of two bounded subsets is studied in some particular cases. A version of the classical Glicksberg's Theorem on the pseudocompactness of the product of two spaces is given in the context of boundedness. This theorem is applied to several particular cases.     

Study of the in-plane and c-axis fluctuation conductivity of melt-textured YBa<Subscript>2</Subscript>Cu<Subscript>3</Subscript>O<Subscript>7</Subscript> under hydrostatic pressure

Strongly confined nano-systems, such as one-dimensional nanowires, feature deviations in their structural, electronic and optical properties from the corresponding bulk. In this work, we investigate the behavior of long-wavelength, optical phonons in vertical arrays of InAs nanowires by Raman spectroscopy. We attribute the main changes in the spectral features to thermal anharmonicity, due to temperature effects, and rule out the contribution of quantum confinement and Fano resonances. We also observe the appearance of surface optical modes, whose details allow for a quantitative, independent estimation of the nanowire diameter. The results shed light onto the mechanisms of lineshape change in low-dimensional InAs nanostructures, and are useful to help tailoring their electronic and vibrational properties for novel functionalities.     

Co-substitution effects on the Fe valence in the BaFe2As2 superconducting compound: a study of hard x-ray absorption spectroscopy

The Fe K x-ray absorption near edge structure of BaFe(2-x)Co(x)As(2) superconductors was investigated. No appreciable alteration in shape or energy position of this edge was observed with Co substitution. This result provides experimental support to previous ab initio calculations in which the extra Co electron is concentrated at the substitute site and do not change the electronic occupation of the Fe ions. Superconductivity may emerge due to bonding modifications induced by the substitute atom that weakens the spin-density-wave ground state by reducing the Fe local moments and/or increasing the elastic energy penalty of the accompanying orthorhombic distortion.     

Baire spaces and Vietoris hyperspaces

We prove that if the Vietoris hyperspace of all nonempty closed subsets of a space is Baire, then all finite powers of must be Baire spaces. In particular, there exists a metrizable Baire space whose Vietoris hyperspace is not Baire. This settles an open problem of R. A. McCoy stated in 1975.

     

Characterizing filters by convergence (with respect to filters) in Banach spaces

Let X be a topological space and let F be a filter on N, recall that a sequence (xn)_n∈N in X is said to be F-convergent to the point x∈X, if for each neighborhood U of x, {n∈N:xn∈U}∈F. By using F-convergence in ?₁ and in Banach spaces, we characterize the P-filters, the P-filters⁺, the weak P-filters, the Q-filters, the Q-filters⁺, the weak Q-filters, the selective filters and the selective⁺ filters.