Dense Subsets of Function Spaces with No Non-Trivial Convergent Sequences

We will show that a monolithic compact space X is not scattered if and only if Cp(X) has a dense subset without non-trivial convergent sequences. Besides, for any cardinal κ ≥ c, the space Rκ has a dense subspace without non-trivial convergent sequences. If X is an uncountable σ-compact space of countable weight, then any dense set Y ⊂ Cp(X) has a dense subspace without non-trivial convergent sequences. We also prove that for any countably compact sequential space X, if Cp(X) has a dense k-subspace, then X is scattered

Some new versions of an old game

summary:The old game is the point-open one discovered independently by F. Galvin [7] and R. Telgársky [17]. Recall that it is played on a topological space $X$ as follows: at the $n$-th move the first player picks a point $x_n\in X$ and the second responds with choosing an open $U_n\ni x_n$. The game stops after $\omega$ moves and the first player wins if $\cup\{U_n:n\in\omega\}=X$. Otherwise the victory is ascribed to the second player. In this paper we introduce and study the games $\theta$ and $\Omega$. In $\theta$ the moves are made exactly as in the point-open game, but the first player wins iff $\cup\{U_n:n\in\omega\}$ is dense in $X$. In the game $\Omega$ the first player also takes a point $x_n\in X$ at his (or her) $n$-th move while the second picks an open $U_n\subset X$ with $x_n\in\overline{U}_n$. The conclusion is the same as in $\theta$, i.e\. the first player wins iff $\cup\{U_n:n\in\omega\}$ is dense in $X$. It is clear that if the first player has a winning strategy on a space $X$ for the game $\theta$ or $\Omega$, then $X$ is in some way similar to a separable space. We study here such spaces $X$ calling them $\theta$-separable and $\Omega$-separable respectively. Examples are given of compact spaces on which neither $\theta$ nor $\Omega$ are determined. It is established that first countable $\theta$-separable (or $\Omega$-separable) spaces are separable. We also prove that \newline 1) all dyadic spaces are $\theta$-separable; \newline 2) all Dugundji spaces as well as all products of separable spaces are $\Omega$-separable; \newline 3) $\Omega$-separability implies the Souslin property while $\theta$-separability does not

When is an ultracomplete space almost locally compact?

[EN] We study spaces X which have a countable outer base in βX; they are called ultracomplete in the most recent terminology. Ultracompleteness implies Cech-completeness and is implied by almost local compactness (≡having all points of non-local compactness inside a compact subset of countable outer character). It turns out that ultracompleteness coincides with almost local compactness in most important classes of isocompact spaces (i.e., in spaces in which every countably compact subspace is compact). We prove that if an isocompact space X is ω-monolithic then any ultracomplete subspace of X is almost locally compact. In particular, any ultracomplete subspace of a compact ω-monolithic space of countable tightness is almost locally compact. Another consequence of this result is that, for any space X such that vX is a Lindelöf Σ-space, a subspace of Cp(X) is ultracomplete if and only if it is almost locally compact. We show that it is consistent with ZFC that not all ultracomplete subspaces of hereditarily separable compact spaces are almost locally compact.Research supported by Consejo Nacional de Ciencia y Tecnología (CONACyT) of Mexico grants 94897 and 400200-5-38164-E.Jardón Arcos, D.; Tkachuk, VV. (2006). When is an ultracomplete space almost locally compact?. Applied General Topology. 7(2):191-201. doi:10.4995/agt.2006.1923.SWORD1912017

Čech-completeness and ultracompleteness in “nice spaces”

summary:We prove that if $X^n$ is a union of $n$ subspaces of pointwise countable type then the space $X$ is of pointwise countable type. If $X^\omega$ is a countable union of ultracomplete spaces, the space $X^\omega$ is ultracomplete. We give, under CH, an example of a Čech-complete, countably compact and non-ultracomplete space, giving thus a partial answer to a question asked in [BY2]

Discrete reflexivity in GO spaces

A property P is discretely reflexive if a space X has P whenever Cl D has P for any discrete set D ⊂ X. We prove that quite a few topological properties are discretely reflexive in GO spaces. In particular, if X is a GO space and Cl D is first countable (paracompact, Lindelöf, sequential or Fréchet-Urysohn) for any discrete D ⊂ X then X is first countable (paracompact, Lindelöf, sequential or Fréchet-Urysoh respectively). We show that a space with a nested local base at every point is discretely locally compact if and only if it is locally compact. Therefore local compactness is discretely reflexive in GO spaces. It is shown that a GO space is scattered if and only if it is discretely scattered. Under CH we show that Čech-completeness is not discretely reflexive even in second countable linearly ordered spaces. However, discrete Čech-completeness of X × X is equivalent to its Čech-completeness if X is a LOTS. We also establish that any discretely Čech-complete Borel set must be Čech-complete

Cofinitely and co-countably projective spaces

We show that X is cofinitely projective if and only if it is a finite union of Alexandroff compactatifications of discrete spaces. We also prove that X is co-countably projective if and only if X admits no disjoint infinite family of uncountable cozero sets. It is shown that a paracompact space X is co-countably projective if and only if there exists a finite set B C X such that B C U ϵ τ (X) implies │X\U│ ≤ ω. In case of existence of such a B we will say that X is concentrated around B. We prove that there exists a space Y which is co-countably projective while there is no finite set B C Y around which Y is concentrated. We show that any metrizable co-countably projective space is countable. An important corollary is that every co-countably projective topological group is countable

Type 2 diabetes and metabolic syndrome: identification of the molecular mechanisms, key signaling pathways and transcription factors aimed to reveal new therapeutical targets

Type 2 diabetes mellitus (T2DM) is a socially important disease with only symptomatic therapy developed due to lack of knowledge about its pathogenesis and underlying mechanism. Insulin resistance (IR) is the first link of T2DM pathogenesis and results in decrease of ability of insulin to stimulate glucose uptake by target cells. Development of IR involves genetic predisposition, excessive nutrition, stress, obesity or chronic inflammation due to disruption of insulin signaling within cells. Molecular mechanisms and markers of IR are characterized rather poorly, which prevents early diagnosis and creation of preventive therapy. Euglycemic clamp test is still a golden standard for IR diagnosis in clinic. Hyperglycemia is a distant consequence of IR in which damaging effect of oxidative and carbonyl stress is realized and diagnosis of T2DM is stipulated. Molecular chaperones and small heat-shock proteins have a protective effect at the early stages of T2DM pathogenesis, preventing development of reticulum stress and apoptosis. Endothelial dysfunction is related to T2DM and its cardiovascular complications, however, it is unknown on which stage of pathogenesis these changes occur and what are their molecular inductors. Finally, transcriptional activity and adipogenic differentiation play an important role in formation of new fat depots from predecessor cells and activation of brown and beige fat demonstrating hypolipidemic and hypoglycemic properties. The aim of this study was investigation of pathophysiological mechanisms of development of IR and endothelial dysfunction, role of transcription factor Prep1 and small heat shock proteins, evaluation of novel methods of diagnostics of IR and therapeutic potential of brown and beige fat, determination of biotargets for new antidiabetic drugs

A nice class extracted from CpC_p-theory

summary:We study systematically a class of spaces introduced by Sokolov and call them Sokolov spaces. Their importance can be seen from the fact that every Corson compact space is a Sokolov space. We show that every Sokolov space is collectionwise normal, $\omega$-stable and $\omega$-monolithic. It is also established that any Sokolov compact space $X$ is Fréchet-Urysohn and the space $C_p(X)$ is Lindelöf. We prove that any Sokolov space with a $G_\delta$-diagonal has a countable network and obtain some cardinality restrictions on subsets of small pseudocharacter lying in $\Sigma$-products of cosmic spaces

A Cp-theory problem book: special features of function spaces

The books in Vladimir Tkachuk’s A Cp-Theory Problem Book series will be the ‘go to’ texts for basic reference to Cp-theory. This second volume, Special Features of Function Spaces, gives a reasonably complete coverage of Cp-theory, systematically introducing each of the major topics and providing  500 carefully selected problems and exercises with complete solutions. Bonus results and open problems are also given. The text is designed to bring a dedicated reader from basic topological principles to the frontiers of modern research covering a wide variety of topics in Cp-theory and general topology at the professional level. The first volume, Topological and Function Spaces © 2011, provided an introduction from scratch to Cp-theory and general topology, preparing the reader for a professional understanding of Cp-theory in the last section of its main text. This second volume continues from the first, and can be used as a textbook for courses in both Cp-theory and general topology as well as a reference guide for specialists working in Cp-theory and related topics.  Additionally, the material can also be considered as an introduction to advanced set theory and descriptive set theory, presenting diverse topics of the theory of function spaces with the topology of pointwise convergence, or Cp-theory which exists at the intersection of topological algebra, functional analysis and general topology. From the Reviews of Topological and Function Spaces: “…It is designed to bring a dedicated reader from the basic topological principles to the frontiers of modern research. Any reasonable course in calculus covers everything needed to understand this book. This volume can also be used as a reference for mathematicians working in or outside the field of topology (functional analysis) wanting to use results or methods of Cp-theory...On the whole, the book provides a useful addition to the literature on Cp-theory, especially at the instructional level." (Mathematical Reviews