Applications of the ergodic iteration theorem

I prove several natural preservation theorems for the countable support iteration. This solves a question of Ros?anowski regarding the preservation of localization properties and greatly simplifies the proofs in the area (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Compact covers and function spaces

For a Tychonoff space X  , we denote by _Cp(X)$C_p(X)$ and _Cc(X)$C_c(X)$ the space of continuous real-valued functions on X equipped with the topology of pointwise convergence and the compact-open topology respectively. Providing a characterization of the Lindelöf Σ-property of X   in terms of _Cp(X)$C_p(X)$, we extend Okunev?s results by showing that if there exists a surjection from _Cp(X)$C_p(X)$ onto _Cp(Y)$C_p(Y)$ (resp. from _Lp(X)$L_p(X)$ onto _Lp(Y)$L_p(Y)$) that takes bounded sequences to bounded sequences, then υY is a Lindelöf Σ-space (respectively K-analytic) if υX has this property. In the second part, applying Christensen?s theorem, we extend Pelant?s result by proving that if X is a separable completely metrizable space and Y   is first countable, and there is a quotient linear map from _Cc(X)$C_c(X)$ onto _Cc(Y)$C_c(Y)$, then Y   is a separable completely metrizable space. We study also a non-separable case, and consider a different approach to the result of J. Baars, J. de Groot, J. Pelant and V. Valov, which is based on the combination of two facts: Complete metrizability is preserved by _?p${?}_p$-equivalence in the class of metric spaces (J. Baars, J. de Groot, J. Pelant). If X   is completely metrizable and _?p${?}_p$-equivalent to a first-countable Y, then Y is metrizable (V. Valov). Some additional results are presented.     

From countable compactness to absolute countable compactness

We show that every countably compact space which is monotonically normal, almost 2-fully normal, radial , or with countable spread is absolutely countably compact. For the first two mentioned properties, we prove more general results not requiring countable compactness. We also prove that every monotonically normal, orthocompact space is finitely fully normal.

     

关于单调亚紧空间

In the first part of this note, we mainly prove that monotone metacompactness is hereditary with respect to closed subspaces and open Fó-subspaces. For a generalized ordered (GO)-space X, we also show that X is monotonically metacompact if and only if its closed linearly ordered extension X* is monotonically metacompact. We also point out that every non-Archimedean space X is monotonically ultraparacompact. In the second part of this note, we give an alternate proof of the result that McAuley space is paracompact and metacompact.     

On R∞ and Q∞-manifolds

We give a characterization of manifolds modeled on $\text{R}{}^{\mathrm{\infty }}\text{= dir lim}$ or $\text{R}{}^{\mathrm{n}}\text{Q}{}^{\mathrm{\infty }}\text{=dir lim Q}{}^{\mathrm{n}}$, where Q is the Hilbert cube, and elementary short proofs of the Open Embedding Theorem for these manifolds and the following theorem generalizing the Stability Theorem: Each fine homotopy equivalence between these manifolds is a near homeomorphism. Moreover we establish the Open Embedding Approximation Theorem.     

On Some Extremal Varieties of Associative Algebras

Suppose that F is a field of prime characteristic p and V _p is the variety of associative algebras over F defined by the identities [[x, y], z] = 0 and x ^p = 0 if p > 2 and by the identities [[x, y], z] = 0 and x ⁴ = 0 if p = 2 (here [x, y] = xy ? yx). As is known, the free algebras of countable rank of the varieties V _p contain non-finitely generated T-spaces. We prove that the varieties V _p are minimal with respect to this property.     

On locally free Abelian groups

The following result is proved in the paper. An Abelian group A is Lw₁, w-equivalent to the free Abelian group of countable rank if and only if it is a countably free Abelian group.Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 121–126.Original Russian Text Copyright © 2005 by E. G. Sklyarenko.This revised version was published online in April 2005 with a corrected issue number.     

The Sequence-covering Compact Images of Paracompact Locally Compact Spaces

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仿紧局部可分空间及其映象

探讨仿紧局部可分空间的一些性质,获得了两个可度量化定理,建立了仿紧局部可分空间的各类序列覆盖L-映象的特征。     

k-COVERS AND CERTAIN QUOTIENT IMAGES OF PARACOMPACT LOCALLY COMPACT SPACES

We continue to discuss some properties of certain k-covers. As applications, we give some characterizations of certain quotient images of paracompact locally compact spaces.