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We characterize coset spaces of topological groups which are coset spaces of (separable) metrizable groups and complete metrizable (Polish) groups. Besides, it is shown that for a $G$-space $X$ with a $d$-open action there is a topological group $H$ of weight and cardinality less than or equal to the weight of $X$ such that $H$ admits a $d$-open action on $X$. This is further applied to show that if $X$ is a separable metrizable coset space then $X$ has a Polish extension which is a coset space of a Polish group.

Near-openly generated groups are introduced. They form a topological and multiplicative subclass of -factorizable groups. Dense and open subgroups, quotients and the Raikov completion of a near-openly generated group are near-openly generated. Almost connected pro-Lie groups, Lindelöf almost metrizable groups and the spaces of all continuous real-valued functions on a Tychonoff space with pointwise convergence topology are near-openly generated. We provide characterizations of near-openly generated groups using methods of inverse spectra and topological game theory. Bibliography: 24 titles.

Вводится подкласс $\mathbb R$-факторизуемых групп - почти открыто порожденные группы. Он является топологическим и мультипликативным. Всюду плотная или открытая подгруппа, факторгруппа почти открыто порожденной группы и ее пополнение по Райкову - почти открыто порожденные группы. Почти связные про-лиевы группы, линделефовы почти метризуемые группы и пространства $C_p(X)$ непрерывных вещественнозначных функций на тихоновских пространствах в топологии поточечной сходимости - почти открыто порожденные группы. Приводятся характеризации почти открыто порожденных групп с использованием методов обратных спектров и теории топологических игр. Библиография: 24 названия.

We prove the analog of Pasynkov's result on finite-dimensionality of topological products for the dimension-like function I defined by S. Iliadis. Let F j be a normal base on a topological space X j , j = 1 , 2 . Then I ( X 1 × X 2 , F 1 ⊗ F 2 ) ≤ φ ( I ( X 1 , F 1 ) , I ( X 2 , F 2 ) ) , where φ is a recursion relation. As a consequence we get the result on the finite-dimensionality of topological products of compacta for the large inductive dimension.

The concept of an R -factorizable G-space is introduced. It is shown that quotients of R -factorizable groups are R -factorizable G-spaces; that the action of an R -factorizable group G on its quotient space X can be extended to an action of the realcompactification vG of G on the realcompactification vX of X; and that vX is G δ -closed in the completion of X with respect to the maximal equiuniformity. The spectral representations of R -factorizable G-spaces are presented.

It is shown that {1,∞} is the set of possible base normal inductive dimensions I of the segment I = [0, 1], and {n, n+1, . . . ,∞} is the set of possible base normal inductive dimensions I of the n-dimensional cubes I n for n ≥ 2.

We present approaches to substitute the acting groups of G-spaces that preserve various properties of actions, such as transitivity, being a coset space, and preserving a fixed equiuniformity in case of a G-Tychonoff space. Such substitutions make it possible to obtain bounds on cardinal invariants of acting groups with admissible topologies, and to investigate metrizability of phase spaces and find their equivariant completions of the same weight.

Using dimension-like invariant base dimension I of a space by a normal base, the inductive dimensions of the product of compacta in Filippov's example are computed.

The presence on a space of an additional algebraic structure, which is compatible with its topology, in many cases imposes strong restrictions on the properties of the space itself. In the paper we discuss connections between spectral characterizations of groups and spaces with their actions when the latter are near to their coset spaces. The compatible system of maps on a space which is induced by a compatible system of maps on the acting group is constructed. This approach allows to give a unified possibility in the study of spaces with special actions of groups with suitable spectral decompositions. Among them are Cech complete groups, their products and subgroups. In particular, it is proved that a pseudocompact coset space of a Cech-complete group is ϰ -metrizable and a compactum which is a coset space of a subgroup of a product of Cech-complete groups is an openly generated compactum. We also show that an arbitrary d -open action of an ℵ 0 -balanced group on a pseudocompact space can be replaced by a d -open action of an ℵ 0 -bounded group.

Topologization of a group of homeomorphisms and its action provide additional possibilities for studying the topological space, the group of homeomorphisms, and their interconnections. The subject of the paper is the use of the property of d-openness of an action (introduced by Ancel under the name of weak micro-transitivity) in the study of spaces with various forms of homogeneity. It is proved that a d-open action of a Cech-complete group is open. A characterization of Polish SLH spaces using d-openness is given, and it is established that any separable metrizable SLH space has an SLH completion that is a Polish space. Furthermore, the completion is realized in coordination with the completion of the acting group with respect to the two-sided uniformity. A sufficient condition is given for extension of a d-open action to the completion of the space with respect to the maximal equiuniformity with preservation of d-openness. A result of van Mill is generalized, namely, it is proved that any homogeneous CDH metrizable compactum is the only G-compactification of the space of rational numbers for the action of some Polish group. Bibliography: 39 titles.

We compared the expression of signal molecules in the culture of human endothelial cells under normal conditions and in atherosclerosis and restenosis. Expression of connexin-37 and sirtuin-1 in atherosclerosis and restenosis surpassed the normal by 2 and 5 times, respectively, and expression of endothelin-1 3-fold surpassed the normal. In restenosis, changes in the expression connexin-37 and endothelin-1 became more pronounced in comparison with atherosclerosis. Connexin-37 and endothelin-1 can serve as predictive markers for prognosis of post-stenting complications in patients with atherosclerosis.

In the present paper three types of covering dimension invariants of a space X are distinguished. Their sets of values are denoted by d-SpU(X), d-SpW(X) and d-Spβ(X). One of the exhibited relations between them shows that the minimal values of d-SpU(X), d-SpW(X) and d-Spβ(X) coincide. This minimal value is equal to the dimension invariant mindim defined by Isbell. We show that if X is a locally compact space, then either d-SpU(X)=[mindimX,∞], or d-SpU(X)=d-Spβ(X)={dimX}. If X is not a pseudocompact space, then [dimX,∞]⊂d-SpU(X); if X is a Lindelöff non-compact space, then d-SpU(X)=[dimX,∞]; if X is a separable metrizable non-compact space, then d-SpW(X)=[mindimX,∞]. Among the properties of covering dimension invariants the generalization of the compactification theorem of Skljarenko is presented. The existence of compact universal spaces in the class of all spaces X with w(X)⩽τ and mindimX⩽n is proved.

The general condition that characterizes the possibility of actionʼs extension over a completion of a phase space is formulated and an example of a G -space that has no Dieudonne complete G -extensions is given. Sufficient conditions (different kinds of rectangular conditions in products) for the actionʼs extensions over the Stone–Cech compactification, the Hewitt realcompactification and the Dieudonne completion of a space are presented. Boundedness, uniform equicontinuity and quasiboundedness of actions are characterized as actionʼs uniform continuity on the (piecewise) semi-uniform product. From this point of view the origin of different examples of actionʼs extensions are explained.

S.D. Iliadis introduced the concept of the dimension-like functions of type dim using the notion of a normal base. Since he considered this notion only from the point of view of the existence of universal spaces in different classes of spaces, in the present paper the basic theorems of dimension theory are obtained for these functions. Their relationship with the relative dimensions of A. Chigogidze and the uniform dimensions of M. Charalambous are shown.

The approach to the problem of the distribution of the functors of the Stone–Cech compactification, the Hewitt realcompactification or the Dieudonne completion with the operation of taking products is discussed using uniform structures on products. In particular, the role of different rectangular conditions is shown. Relative analogues of this question and new examples of (strongly) rectangular products are presented. Characterizations of bounded rectangular subsets of the product are given.

The presence of an algebraic structure on a space, which is compatible with its topology, in many cases imposes very strong restrictions on the properties of the space itself. Conditions are found which must be satisfied by the actions in order for the phase space to be a d-space (Dugundji compactum). This investigation allows the range of G-spaces that are d-spaces (Dugundji compacta) to be substantially widened. It is shown that all the cases known to the authors where a G-space (a topological group, one of its quotient spaces) is a d-space can be realized using equivariant maps. Bibliography: 39 titles.

Properties of the large inductive dimension of a space by its normal base introduced by S. Iliadis are studied. The proposed dimension-like functions generalize both the classic dimensions Ind, Ind0 and the relative inductive dimensions I. Properties of the normal base characterizing the fulfillment of basic classic theorems of the dimension theory (sum, subset and product theorems) are found.

In this paper we characterize closed subsets of products of simple compact spaces with noncoinciding dimensions dim and ind.

The first part of the paper is concerned with conditions under which the inequality dim X × Y ≤ dim X + dim Y and similar inequalities for infinite topological products hold. The second part contains examples of spaces such that the sums of their dimensions are smaller than the dimensions of their products.

The semilattice of G-compactifications of a G-Tikhonov space X is studied. The question of what sets containing X and contained in the maximal G-compactification s G X must be contained also in all other G-compactifications of X is considered. Conditions for s G X to be the completion of the G-space X with respect to a natural uniformity (proximity) on X are obtained. Sufficient conditions for the existence of a smallest (minimal, unique) G-compactification of X are given. © 2005 Springer Science+Business Media, Inc.

Publisher Summary This chapter discusses some questions on topological algebra. The chapter begins with the concept of unconditionally closed and algebraic sets. Markov called a subset A of a group G unconditionally closed in G if it is closed in any Hausdorff group topology on G . Clearly, all solution sets of equations in G , as well as their finite unions and arbitrary intersections, are unconditionally closed. A subset A of a group G with identity element 1 is said to be elementary algebraic in G if there exists a word w = w ( x ) in the alphabet G U { x ± 1 } ( x is a variable) such that A = { x ∈ G : w ( x ) = 1}. Finite unions of elementary algebraic sets are additively algebraic sets. An arbitrary intersection of additively algebraic sets is called algebraic. Thus, the algebraic sets in G are the solution sets of arbitrary conjunctions of finite disjunctions of equations. The concepts related to dimensions of metrizable groups, values of metrics, and free groups are also discussed in the chapter. Details of Maltsev spaces and retracts of groups are also presented in the chapter.