We use topological consequences of PFA, MA$_{ω_1}$(S)[S] and PFA(S)[S] proved by other authors to show that normal first countable linearly H-closed spaces with various additionals properties are compact in these models.
We answer a question of Yasui. Morever, we show that if a Tychonoff space Y is countably 1-paracompact in every Tychonoff space X that contains Y as a closed subspace then Y is linearly Lindelof.
A. V. Arhangel'skiĭ introduced in 2012, when he was visiting the department of Mathematics at King Abduaziz University, new weaker versions of normality, called \it $C$-normality, \rm and \it countable normality. \rm The purpose of this paper is to investigate countable normality property. We prove that normality implies countable normality but the converse is not true in general. We present some examples to show relationships between countable normality and other weaker versions of normality such as $C$-normality, $L$-noramlity, and mild normality. We answer the following open problem of Arha…
N.N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences（俄罗斯科学院乌拉尔分校 N.N.克拉索夫斯基数学力学研究所）；Department of General Topology and Geometry, Mechanics and Mathematics Faculty, M. V. Lomonosov Moscow State University（莫斯科国立大学机械数学系一般拓扑与几何教研室）
We call a space $X$ {\it weakly linearly Lindelöf} if for any family $\mathcal{U}$ of non-empty open subsets of $X$ of regular uncountable cardinality $κ$, there exists a point $x\in X$ such that every neighborhood of $x$ meets $κ$-many elements of $\mathcal{U}$. We also introduce the concept of {\it almost discretely Lindelöf} spaces as the ones in which every discrete subspace can be covered by a Lindelöf subspace. We prove that, in addition to linearly Lindelöf spaces, both weakly Lindelöf spaces and almost discretely Lindelöf spaces are weakly linearly Lindelöf. The main result of the pape…
We prove that the countable product of supercomplete spaces having a countable closed cover consisting of partition-complete subspaces is supercomplete with respect to its metric-fine coreflection. Thus, countable products of sigma-partition-complete paracompact spaces are again paracompact. On the other hand, we show that in arbitrary products of partition-complete paracompact spaces, all normal covers can be obtained via the locally fine coreflection of the product of fine uniformities.
We show that a Tychonoff discretely star-Lindelof space can have arbitrarily big extent and note that there are consistent examples of normal discretely star-Lindelof spaces with uncountable extent.
A topological space $X$ is called almost discretely Lindelöf if every discrete set $D \subset X$ is included in a Lindelöf subspace of $X$. We say that the space $X$ is {\em $μ$-sequential} if for every non-closed set $A \subset X$ there is a sequence of length $\le μ$ in $A$ that converges to a point which is not in $A$. With the help of a technical theorem that involves elementary submodels, we establish the following two results concerning such spaces. (1) For every almost discretely Lindelöf $T_3$ space $X$ we have $|X| \le 2^{χ(X)}$. (2) If $X$ is a $μ$-sequential $T_2$ space of pseudocha…
Under $(\diamondsuit^*)$ we construct a locally countable, locally compact, 0-dimensional $T_2$ space $X$ of size $\omg$ which is aD however not even linearly D. This consistently answers a question of Arhangel'skii, whether aD implies D. Furthermore we answer two problems concerning characterization of linearly D-spaces, raised by Guo and Junnila.
In applications it is useful to know whether a topological preordered space is normally preordered. It is proved that every $k_ω$-space equipped with a closed preorder is a normally preordered space. Furthermore, it is proved that second countable regularly preordered spaces are perfectly normally preordered and admit a countable utility representation.
In this paper, the class of all linearly ordered topological spaces (LOTS) quasi-ordered by the embeddability relation is investigated. In ZFC it is proved that for countable LOTS this quasi-order has both a maximal (universal) element and a finite basis. For the class of uncountable LOTS of cardinality $κ$ it is proved that this quasi-order has no maximal element for $κ$ at least the size of the continuum and that in fact the dominating number for such quasi-orders is maximal, i.e. $2^κ$. Certain subclasses of LOTS, such as the separable LOTS, are studied with respect to the top and internal …
A space is called linearly H-closed iff any chain cover possesses a dense member. This property lies strictly between feeble compactness and H-closedness. While regular H-closed spaces are compact, there are linearly H-closed spaces which are even collectionwise normal and Fréchet-Urysohn. We give examples in other classes, and ask whether there is a first countable normal linearly H-closed non-compact space in ZFC. We show that PFA implies a negative answer if the space is moreover either locally separable or locally compact and locally ccc. Ostaszewski space (built with $\diamondsuit$) is an…
Tkachuk and Wilson proved that a regular first countable cellular-compact space has cardinality not exceeding the continuum. In the same paper they asked if this result continues to hold for Hausdorff spaces. Xuan and Song considered the same notion and asked if every cellular-compact space is weakly Lindelof. We answer the last question for first countable spaces. As a byproduct of this result, we present a somewhat different proof of Tkachuk and Wilson theorem, valid for the wider class of Urysohn spaces. The result actually holds for a class of spaces in between cellular-compact and cellula…
A space $X$ is said to be "cellular-Lindelöf" if for every cellular family $\mathcal{U}$ there is a Lindelöf subspace $L$ of $X$ which meets every element of $\mathcal{U}$. Cellular-Lindelöf spaces generalize both Lindelöf spaces and spaces with the countable chain condition. Solving questions of Xuan and Song, we prove that every cellular-Lindelöf monotonically normal space is Lindelöf and that every cellular-Lindelöf space with a regular $G_δ$-diagonal has cardinality at most $2^\mathfrak{c}$. We also prove that every normal cellular-Lindelöf first-countable space has cardinality at most con…
The countable uniform power (or uniform box product) of a uniform space $X$ is a special topology on ${}^ωX$ that lies between the Tychonoff topology and the box topology. We solve an open problem posed by P. Nyikos showing that if $X$ is a compact proximal space then the countable uniform power of $X$ is also proximal (although it is not compact). By recent results of J. R. Bell and G. Gruenhage this implies that the countable uniform power of a Corson compactum is collectionwise normal, countably paracompact and Fréchet-Urysohn. We also give some results about first countability, realcompact…
We work in set-theory without choice $\ZF$. Given a closed subset $F$ of $[0,1]^I$ which is a bounded subset of $\ell^1(I)$ ({\em resp.} such that $F \subseteq \ell^0(I)$), we show that the countable axiom of choice for finite subsets of $I$, ({\em resp.} the countable axiom of choice $\ACD$) implies that $F$ is compact. This enhances previous results where $\ACD$ ({\em resp.} the axiom of Dependent Choices $\DC$) was required. Moreover, if $I$ is linearly orderable (for example $I=\IR$), the closed unit ball of $\ell^2(I)$ is weakly compact (in $\ZF$).
We describe those complete linearly ordered topological spaces $X$ which are homogeneous (=CHLOTS). That is, $X$ is order isomorphic with any nonempty open interval in $X$. Using countable tail-like ordinals as indices, we build towers of distinct CHLOTS. Using tree constructions we are able to extend the towers and to describe an inductive procedure which yields every CHLOTS.
In this survey, my aim has been to discuss the use of sequences and countable sets in general topology. In this way I have been led to consider five different classes of topological spaces: first countable spaces, sequential spaces, Frechet spaces, spaces of countable tightness and perfect spaces. We are going to look at how these classes are related, and how well the various properties behave under certain operations, such as taking subspaces, products, and images under proper mappings. Where they are not well behaved we take the opportunity to consider some relevant examples, which are often…
We continue in this paper the study of locally minimal groups started in \cite{LocMin}. The minimality criterion for dense subgroups of compact groups is extended to local minimality. Using this criterion we characterize the compact abelian groups containing dense countable locally minimal subgroups, as well as those containing dense locally minimal subgroups of countable free-rank. We also characterize the compact abelian groups whose torsion part is dense and locally minimal. We call a topological group $G$ {\it almost minimal} if it has a closed, minimal normal subgroup $N$ such that the qu…
We derive many upper bounds on the submetrizability number and $i$-weight of paratopological groups and topological monoids with open shifts. In particular, we prove that each first countable Hausdorff paratopological group is submetrizable thus answering a problem of Arhangelskii posed in 2002. Also we construct an example of a zero-dimensional (and hence regular) Hausdorff paratopological abelian group $G$ with countable pseudocharacter which is not submetrizable. In fact, all results on the $i$-weight and submetrizability are derived from more general results concerning normally quasi-unifo…
In this paper, two outwardly different graphs, namely, the zero divisor graph $Γ(C_c(X))$ and the comaximal graph $Γ_2^{'}(C_c(X))$ of the ring $C_c(X)$ of all real-valued continuous functions having countable range, defined on any Hausdorff zero dimensional space $X$, are investigated. It is observed that these two graphs exhibit resemblance, so far as the diameters, girths, connectedness, triangulatedness or hypertriangulatedness. are concerned. However, the study reveals that the zero divisor graph $Γ(A_c(X))$ of an intermediate ring $A_c(X)$ of $C_c(X)$ is complemented if and only if the s…
We study linearly ordered spaces which are Valdivia compact in their order topology. We find an internal characterization of these spaces and we present a counter-example disproving a conjecture posed earlier by the first author. The conjecture asserted that a compact line is Valdivia compact if its weight does not exceed aleph one, every point of uncountable character is isolated from one side and every closed first countable subspace is metrizable. It turns out that the last condition is not sufficient. On the other hand, we show that the conjecture is valid if the closure of the set of poin…
Let $Y$ be a metrizable space containing at least two points, and let $X$ be a $Y_{\mathcal{I}}$-Tychonoff space for some ideal $\mathcal{I}$ of compact sets of $X$. Denote by $C_{\mathcal{I}}(X,Y)$ the space of continuous functions from $X$ to $Y$ endowed with the $\mathcal{I}$-open topology. We prove that $C_{\mathcal{I}}(X,Y)$ is Fréchet - Urysohn iff $X$ has the property $γ_{\mathcal{I}}$. We characterize zero - dimensional Tychonoff spaces $X$ for which the space $C_{\mathcal{I}}(X,{\bf 2})$ is sequential. Extending the classical theorems of Gerlits, Nagy and Pytkeev we show that if $Y$ i…
