Compact spaces generated by retractions

We study compact spaces which are obtained from metric compacta by iterating the operation of inverse limit of continuous sequences of retractions. This class, denoted by R, has been introduced in [M. Burke, W. Kubi?, S. Todor?evi?, Kadec norms on spaces of continuous functions, http://arxiv.org/abs/math.FA/0312013]. Allowing continuous images in the definition of class R, one obtains a strictly larger class, which we denote by RC. We show that every space in class RC is either Corson compact or else contains a copy of the ordinal segment ω₁+1. This improves a result of Kalenda from [O. Kalenda, Embedding of the ordinal segment [0,ω₁] into continuous images of Valdivia compacta, Comment. Math. Univ. Carolin. 40 (4) (1999) 777-783], where the same was proved for the class of continuous images of Valdivia compacta. We prove that spaces in class R do not contain cutting P-points (see the definition below), which provides a tool for finding spaces in RC?R. Finally, we study linearly ordered spaces in class RC. We prove that scattered linearly ordered compacta belong to RC and we characterize those ones which belong to R. We show that there are only 5 types (up to order isomorphism) of connected linearly ordered spaces in class R and all of them are Valdivia compact. Finally, we find a universal pre-image for the class of all linearly ordered Valdivia compacta.     

The structure of Valdivia compact lines

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 ℵ₁, 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 points of uncountable character is scattered. This improves an earlier result of the first author.     

Linearly ordered compacta and Banach spaces with a projectional resolution of the identity

We construct a compact linearly ordered space Kω₁ of weight ℵ₁, such that the space C(Kω₁) is not isomorphic to a Banach space with a projectional resolution of the identity, while on the other hand, Kω₁ is a continuous image of a Valdivia compact and every separable subspace of C(Kω₁) is contained in a 1-complemented separable subspace. This answers two questions due to O. Kalenda and V. Montesinos.     

Fodor-type Reflection Principle and reflection of metrizability and meta-Lindelöfness

We introduce a new reflection principle which we call “Fodor-type Reflection Principle” (FRP). This principle follows from but is strictly weaker than Fleissner's Axiom R. For instance, FRP does not impose any restriction on the size of the continuum, while Axiom R implies that the continuum has size ?ℵ₂.We show that FRP implies that every locally separable countably tight topological space X is meta-Lindelöf if all of its subspaces of cardinality ?ℵ₁ are (Theorem 4.3). It follows that, under FRP, every locally (countably) compact space is metrizable if all of its subspaces of cardinality ?ℵ₁ are (Corollary 4.4). This improves a result of Balogh who proved the same assertion under Axiom R.We also give several other results in this vein, some in ZFC, others in some further extension of ZFC. For example, we prove in ZFC that if X is a locally (countably) compact space of singular cardinality in which every subspace of smaller size is metrizable then X itself is also metrizable (Corollary 5.2).     

On some classes of Lindelöf Σ-spaces

We consider special subclasses of the class of Lindelöf Σ-spaces obtained by imposing restrictions on the weight of the elements of compact covers that admit countable networks: A space X is in the class LΣ(?κ) if it admits a cover by compact subspaces of weight κ and a countable network for the cover. We restrict our attention to κ?ω. In the case κ=ω, the class includes the class of metrizably fibered spaces considered by Tkachuk, and the P-approximable spaces considered by Tka?enko. The case κ=1 corresponds to the spaces of countable network weight, but even the case κ=2 gives rise to a nontrivial class of spaces. The relation of known classes of compact spaces to these classes is considered. It is shown that not every Corson compact of weight ℵ₁ is in the class LΣ(?ω), answering a question of Tkachuk. As well, we study whether certain compact spaces in LΣ(?ω) have dense metrizable subspaces, partially answering a question of Tka?enko. Other interesting results and examples are obtained, and we conclude the paper with a number of open questions.     

Indestructibility of compact spaces

In this article we investigate which compact spaces remain compact under countably closed forcing. We prove that, assuming the Continuum Hypothesis, the natural generalizations to _ω1${\omega }_1$-sequences of the selection principle and topological game versions of the Rothberger property are not equivalent, even for compact spaces. We also show that Tall and Usuba?s “_ℵ1${\aleph }_1$-Borel Conjecture” is equiconsistent with the existence of an inaccessible cardinal.     

On nonnormal products and a set of A.H. Stone

A variant of Michael's example is given to the following effect: there is a Lindelöf space M of weight ℵ₁, with all Gδ-sets open, such that M×B(ℵ₁) is nonnormal. This answers a question from [K. Alster, On the class of ω₁-metrizable spaces whose product with every paracompact space is paracompact, Topology Appl. 153 (2006) 2508-2517].     

A first countable linearly Lindelöf not Lindelöf topological space

A topological space X is called linearly Lindelöf if every increasing open cover of X has a countable subcover. It is well known that every Lindelöf space is linearly Lindelöf. The converse implication holds only in particular cases, such as X being countably paracompact or if nw(X)<ℵω.Arhangel?skii and Buzyakova proved that the cardinality of a first countable linearly Lindelöf space does not exceed ℵ₀². Consequently, a first countable linearly Lindelöf space is Lindelöf if ℵω>ℵ₀². They asked whether every linearly Lindelöf first countable space is Lindelöf in ZFC. This question is supported by the fact that all known linearly Lindelöf not Lindelöf spaces are of character at least ℵω. We answer this question in the negative by constructing a counterexample from MA+ℵω<ℵ₀².A modification of Alster?s Michael space that is first countable is presented.     

On topological spaces of singular density and minimal weight

We introduce a weakening of the generalized continuum hypothesis, which we will refer to as the prevalent singular cardinals hypothesis, and show it implies that every topological space of density and weight ℵω₁ is not hereditarily Lindelöf.The assumption PSH is very weak, and in fact holds in all currently known models of ZFC.     

Weak covering properties and selection principles

No convenient internal characterization of spaces that are productively Lindelöf is known. Perhaps the best general result known is Alster?s internal characterization, under the Continuum Hypothesis, of productively Lindelöf spaces which have a basis of cardinality at most _ℵ1${\aleph }_1$. It turns out that topological spaces having Alster?s property are also productively weakly Lindelöf. The weakly Lindelöf spaces form a much larger class of spaces than the Lindelöf spaces. In many instances spaces having Alster?s property satisfy a seemingly stronger version of Alster?s property and consequently are productively X, where X is a covering property stronger than the Lindelöf property. This paper examines the question: When is it the case that a space that is productively X is also productively Y, where X and Y are covering properties related to the Lindelöf property.     

Remarks on countability and star covering properties

In this paper, we construct an example of a T₄ feebly Lindelöf space X which is not star Lindelöf under ℵ₀²=ℵ₁², which gives a partial answer to Alas, Junqueira and Wilson (2011) [1, Question 4].     

Almost isometric embeddings into configuration-compacta

We show that a Polish metric space X which has ‘compact configuration spaces’ is free of almost isometric embeddings, i.e. given such an embedding into X one gets an isometric embedding. By assuming a continuous version of ℵ₀-homogeneity we get the converse statement, and also prove almost isometry uniqueness.     

Valdivia compact groups are products

It is shown that every Valdivia compact group is homeomorphic to a product of metrizable compacta.     

More on perfectly normal non-realcompact spaces

We use the space associated with a guessing sequence on ω₁ to show that it is consistent with CH that there exists a locally countable, first-countable, locally compact, perfectly normal, non-realcompact space of size ℵ₁ which does not contain any sub-Ostaszewski spaces. By a similar technique, it is shown to be consistent with that there exists a locally countable, first-countable, perfectly normal, non-realcompact space of size ℵ₁.     

Dissipated compacta

The dissipated spaces form a class of compacta which contains both the scattered compacta and the compact LOTSes (linearly ordered topological spaces), and a number of theorems true for these latter two classes are true more generally for the dissipated spaces. For example, every regular Borel measure on a dissipated space is separable.The standard Fedor?uk S-space (constructed under ?) is dissipated. A dissipated compact L-space exists iff there is a Suslin line.A product of two compact LOTSes is usually not dissipated, but it may satisfy a weakening of that property. In fact, the degree of dissipation of a space can be used to distinguish topologically a product of n LOTSes from a product of m LOTSes.     

Spaces with large weak extent

In this paper, we show the following statements:

(1)

For any cardinal κ, there exists a pseudocompact centered-Lindelöf Tychonoff space X such that we(X)?κ.

(2)

Assuming ℵ₀²=ℵ₁², there exists a centered-Lindelöf normal space X such that we(X)?ω₁.

     

Free Boolean algebras over unions of two well orderings

Given a partially ordered set P there exists the most general Boolean algebra which contains P as a generating set, called the free Boolean algebra over P. We study free Boolean algebras over posets of the form P=P₀∪P₁, where P₀, P₁ are well orderings. We call them nearly ordinal algebras.Answering a question of Maurice Pouzet, we show that for every uncountable cardinal κ there are κ² pairwise non-isomorphic nearly ordinal algebras of cardinality κ.Topologically, free Boolean algebras over posets correspond to compact 0-dimensional distributive lattices. In this context, we classify all closed sublattices of the product (ω₁+1)×(ω₁+1), showing that there are only ℵ₁ many types. In contrast with the last result, we show that there are ℵ₁² topological types of closed subsets of the Tikhonov plank (ω₁+1)×(ω+1).     

On the Steinhaus property in topological groups

Let G be a locally compact Abelian group and μ a Haar measure on G. We prove: (a) If G is connected, then the complement of a union of finitely many translates of subgroups of G with infinite index is μ-thick and everywhere of second category. (b) Under a simple (and fairly general) assumption on G, for every cardinal number m such that ℵ₀?m?|G| there is a subgroup of G of index m that is μ-thick and everywhere of second category. These results extend theorems by Muthuvel and Erd?s-Marcus, respectively. (b) also implies a recent theorem by Comfort-Raczkowski-Trigos stating that every nondiscrete compact Abelian group G admits ²|G|-many μ-nonmeasurable dense subgroups.     

Homotopical properties of upper semifinite hyperspaces of compacta

In this paper we study homotopical properties of a special neighborhood system, which is denoted by {Uε}_?>0, for the canonical embedding of a compact metric space in its upper semifinite hyperspace to get results in the shape theory for compacta. We also point out that there are spaces with the shape of finite discrete spaces and having not the homotopy type of any T₁-space