Vladimir V. Tkachuk

Irresolvable and submaximal spaces: Homogeneity versus σ-discreteness and new ZFC examples1

Abstract An example of an irresolvable dense subspace of {0,1} c is constructed in ZFC. We prove that there can be no dense maximal subspace in a product of first countable spaces, while under Booths Lemma there exists a dense submaximal subspace in [0,1] c . It is established that under the axiom of constructibility any submaximal Hausdorff space is σ-discrete. Hence it is consistent that there are no submaximal normal connected spaces. If there exists a measurable cardinal, then there are models of ZFC with non-σ-discrete maximal spaces. We prove that any homogeneous irresolvable space of non-measurable cardinality is of first category. In particular, any homogeneous submaximal space is strongly σ-discrete if there are no measurable cardinals.

On the extent of star countable spaces

For a topological property P, we say that a space X is star Pif for every open cover Uof the space X there exists Y ⊂ X such that St(Y,U) = X and Y has P. We consider star countable and star Lindelöf spaces establishing, among other things, that there exists first countable pseudocompact spaces which are not star Lindelöf. We also describe some classes of spaces in which star countability is equivalent to countable extent and show that a star countable space with a dense σ-compact subspace can have arbitrary extent. It is proved that for any ω1-monolithic compact space X, if Cp(X)is star countable then it is Lindelöf.

Topological groups with thin generating sets

Abstract A discrete subset S of a topological group G with identity 1 is called suitable for G if S generates a dense subgroup of G and S ∪ {1} is closed in G. We study various algebraic and topological conditions on a group G which imply the existence of a suitable set for G as well as the restraints imposed by the existence of such a set. The classes S c , S g and S cg of topological groups having a closed, generating and a closed generating suitable set are considered. The problem of stability of these classes under the product, direct sum operations and taking subgroups or quotients is investigated. We show that (totally) minimal Abelian groups often have a suitable set. It is also proved that every Abelian group endowed with the finest totally bounded group topology has a closed generating suitable set. More generally, the Bohr topology of every locally compact Abelian group admits a suitable set.

Connectifying some spaces

Abstract A Hausdorff space X is called (countably) connectifiable if there exists a connected Hausdorff space Y (with |Y⊮X| ⩽ ω ; respectively) such that X embeds densely into Y . We prove that it is consistent with ZFC that there exists a regular dense in itself countable space which is not countably connectifiable giving thus a partial answer to Problem 3.9 of Watson and Wilson (1993). On the other hand we show that Martins axiom implies that every countable dense in itself space X with πω ( X ) ω is countably connectifiable. We also establish that a separable metrizable space without open compact subsets can be densely embedded in a metric continuum.

Some topological groups with and some without suitable sets

Abstract If a discrete subset S of a topological group G with identity 1 generates a dense subgroup of G and S∪{1} is closed in G , then S is called a suitable set for G . We construct in ZFC a Lindelof topological group L such that t(L)·ψ(L)≤ℵ 0 and L does not have a suitable set. We also give a ZFC example of a countably compact topological group H with no suitable set; in addition, the closure of every countable subset of H is compact. It is proved that a non-pseudocompact topological group with a dense strictly σ -discrete subset has a closed suitable set. This implies, in particular, that a free (Abelian) topological group on a metrizable space has a closed suitable set.

Pseudocompact Whyburn spaces need not be Fréchet

We prove in ZFC that there exists a Tychonoff pseudocompact scattered AP-space of uncountable tightness. We give some sufficient and necessary conditions for a P-space to be AP as well as a characterization of AP-property in linearly ordered topological spaces.

Behaviour of the Lindelöf Σ-property in iterated function spaces

Abstract We prove that if C p ( X ) is a Lindelof Σ -space, then C p ,2 n +1 ( X ) is a Lindelof Σ -space for every natural n. As a consequence, it is established that υC p C p ( X ) has the Lindelof Σ -property. This answers Problem 47 of Arhangelskii (Recent Progress in General Topology, Elsevier Science, 1992). Another consequence is that only the following distribution of the Lindelof Σ -property is possible in iterated function spaces: (1) C p , n +1 ( X ) is a Lindelof Σ -space for every n ∈ ω ; (2) C p , n +1 ( X ) is a Lindelof Σ -space only for odd n ∈ ω ; (3) C p , n +1 ( X ) is a Lindelof Σ -space only for even n ∈ ω ; (4) for any n ∈ ω the space C p , n +1 ( X ) is not a Lindelof Σ -space. As an application of the developed technique, we prove that, if X is a Tychonoff space such that ω 1 is a caliber for X and C p ( X ) is a Lindelof Σ -space, then X has a countable network. This settles Problem 69 of Arhangelskii (Recent Progress in General Topology, Elsevier Science, 1992).

Lindelöf Σ-spaces: an omnipresent class

This survey’s first object is to introduce the reader to Lindelöf Σ-spaces; since the author would like this introduction to be useful for postgraduate students and non-specialists in the area, most of the basic results are given with complete proofs.The second object is to make an overview of the recent progress achieved in the study of Lindelöf Σ-spaces. Several popular topics are presented together with open problems, some old and some new. The main idea is to show the areas of major activities displaying what is being done nowadays and trying to outline the trends of their future development. A big percentage of the cited results and problems are new, i.e., published/obtained in the 21-st century. However, some classical old theorems and questions are also discussed the author being convinced that they are worth to be repeated with a new emphasis due to the modern vision of the area.ResumenEl primer objetivo de este artículo es brindar una introducción a la teoría de los espacios Lindelöf Σ; el autor quisiera que dicha introducción fuera útil tanto para los estudiantes de posgrado como para los que no son especialistas en el área, así que la mayoría de los resultados básicos se presentan con demostraciones completas.El segundo objetivo es describir, a grandes rasgos, los avances modernos en el estudio de los espacios Lindelöf Σ. Al respecto presentamos algunos temas populares, tanto recientes como ya establecidos desde hace tiempo. La idea principal es mostrar las áreas de mayor actividad, esbozando lo que se está haciendo hoy en día y tratando de visualizar las tendencias para el futuro desarrollo de dichas áreas. Un porcentaje considerable de los resultados citados son nuevos, es decir, obtenidos en el siglo 21. Sin embargo, presentamos también bastantes teoremas clásicos y preguntas abiertas viejas ya que el autor está convencido de que merecen ser mencionados con un nuevo énfasis debido a la visión moderna del área.

The Approximation of Compacta by Finite T0-Spaces

It has long been known that compact Hausdorff spaces can be approximated using finite T 0-spaces, and that many can be represented as inverse limits of polyhedra. Here we study the relationship between these two types of representation. In Section 4, we define the concept of a calming map and show that the Hausdorff reflection of the limit of an inverse sequence of finite T 0-spaces and calming maps is the inverse limit of their corresponding polytopes and piecewise linear maps. Thus each k -dimensional metric compactum (respectively, continuum) can be characterized as the Hausdorff reflection of the limit of an inverse sequence with calming bonding maps of finite (respectively, connected) T 0-spaces whose dimension is k; an infinite-dimensional version of this is also found.

Amsterdam properties of Cp(X) imply discreteness of X

Weprove, among otherthings, thatif C p(X)issubcompactin thesenseofde Groot, then the space X is discrete. This generalizes a series of previous results on completeness properties of function spaces.