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.     

Extending compact topologies to compact Hausdorff topologies in ZF

Within the framework of Zermelo-Fraenkel set theory ZF, we investigate the set-theoretical strength of the following statements:

(1)

For every family(Ai)_i∈Iof sets there exists a family(Ti)_i∈Isuch that for everyi∈I(Ai,Ti)is a compactT₂space.

(2)

For every family(Ai)_i∈Iof sets there exists a family(Ti)_i∈Isuch that for everyi∈I(Ai,Ti)is a compact, scattered, T₂space.

(3)

For every set X, every compactR₁topology (itsT₀-reflection isT₂) on X can be enlarged to a compactT₂topology.

We show:

(a)

(1) implies every infinite set can be split into two infinite sets.

(b)

(2) iff AC.

(c)

(3) and “there exists a free ultrafilter” iff AC.

We also show that if the topology of certain compact T₁ spaces can be enlarged to a compact T₂ topology then (1) holds true. But in general, compact T₁ topologies do not extend to compact T₂ ones.     

Locally compact spaces of countable core and Alexandroff compactification

We introduce a new cardinal invariant, core of a space, defined for any locally compact Hausdorff space X and denoted by cor(X). Locally compact spaces of countable core generalize locally compact σ-compact spaces in a way that is slightly exotic, but still quite natural. We show in Section 1 that under a broad range of conditions locally compact spaces of countable core must be σ-compact. In particular, normal locally compact spaces of countable core and realcompact locally compact spaces of countable core are σ-compact. Perfect mappings preserve the class of spaces of countable core in both directions (Section 2). The Alexandroff compactification aX is weakly first countable at the Alexandroff point a if and only if cor(X)=ω (Section 3). Two examples of non-σ-compact locally compact spaces of countable core are discussed in Section 3. We also extend the well-known theorem of Alexandroff and Urysohn on the cardinality of perfectly normal compacta to compacta satisfying a weak version of perfect normality. Several open problems are formulated.     

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.     

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     

Small Valdivia compact spaces

We prove a preservation theorem for the class of Valdivia compact spaces, which involves inverse sequences of retractions of a certain kind. Consequently, a compact space of weight?ℵ₁ is Valdivia compact iff it is the limit of an inverse sequence of metric compacta whose bonding maps are retractions. As a corollary, we show that the class of Valdivia compacta of weight?ℵ₁ is preserved both under retractions and under open 0-dimensional images. Finally, we characterize the class of all Valdivia compacta in the language of category theory, which implies that this class is preserved under all continuous weight preserving functors.     

Products of straight spaces

A metric space X is straight if for each finite cover of X by closed sets, and for each real valued function f on X, if f is uniformly continuous on each set of the cover, then f is uniformly continuous on the whole of X. A locally connected space is straight iff it is uniformly locally connected (ULC). It is easily seen that ULC spaces are stable under finite products. On the other hand the product of two straight spaces is not necessarily straight. We prove that the product X×Y of two metric spaces is straight if and only if both X and Y are straight and one of the following conditions holds:

(a)

both X and Y are precompact;

(b)

both X and Y are locally connected;

(c)

one of the spaces is both precompact and locally connected.

In particular, when X satisfies (c), the product X×Z is straight for every straight space Z.Finally, we characterize when infinite products of metric spaces are ULC and we completely solve the problem of straightness of infinite products of ULC spaces.     

Observations on quasi-uniform products

We prove that any product of quotient maps in the category of quasi-uniform spaces and quasi-uniformly continuous maps is a quotient map. We also show that a quasi-uniformly continuous map from a product of quasi-uniform spaces into a quasi-pseudometric T₀-space depends on countably many coordinates.Furthermore we characterize those quasi-uniformities that are unique in their quasi-proximity class and prove that this property is preserved under arbitrary products in the category of quasi-uniform spaces.     

Fréchet-Urysohn for finite sets, II

We continue our study [G. Gruenhage, P.J. Szeptycki, Fréchet Urysohn for finite sets, Topology Appl. 151 (2005) 238-259] of several variants of the property of the title. We answer a question from that paper by showing that a space defined in a natural way from a certain Hausdorff gap is a Fréchet α₂ space which is not Fréchet-Urysohn for 2-point sets (FU₂), and answer a question of Hrušák by showing that under MAω₁, no such “gap space” is FU₂. We also introduce versions of the properties which are defined in terms of “selection principles”, give examples when possible showing that the properties are distinct, and discuss relationships of these properties to convergence in product spaces, to the αi-spaces of A.V. Arhangel'skii, and to topological games.     

Separation properties in algebraic categories of topological spaces

Full subcategories C ? Top of the category of topological spaces, which are algebraic over Set in the sense of Herrlich [2], have pleasant separation properties, mostly subject to additional closedness assumptions. For instance, every C-object is a T₁-space, if the two-element discrete space belongs to C. Moreover, if C is closed under the formation of finite powers in Top and even varietal [2], then every C-object is Hausdorff. Hence, the T₂-axiom turns out to be (nearly) superfluous in Herrlich's and Strecker's characterization of the category of compact Hausdorff spaces [1], although it is essential for the proof.If we think of C-objects X as universal algebras (with possibly infinite operations), then the subalgebras of X form the closed sets of a compact topology on X, provided that the ordinal spaces [0, β] belong to C. This generalizes a result in [3]. The subalgebra topology is used to prove criterions for the Hausdorffness of every space in C, if C is only algebraic.     

On sequentially compact T2 compactifications

It is known that a compact space can fail to be sequentially compact. In this paper we consider the following problem: when does a space admit a sequentially compact T₂ compactification? In the first section we develop a method to produce such compactifications, and we apply it in the second section to study the question using coverings.Moreover, we obtain solutions for locally compact T₂ spaces, and for metrizable spaces.     

Selection principles in hyperspaces with generalized Vietoris topologies

We continue investigations of ?ech closure spaces and their hyperspaces started in [M. Mrševi?, M. Jeli?, Selection principles and hyperspace topologies in closure spaces, J. Korean Math. Soc. 43 (2006) 1099-1114] and [M. Mrševi?, M. Jeli?, Selection principles, γ-sets and αi-properties in ?ech closure spaces, Topology Appl., in press], focusing on generalized upper and lower Vietoris topologies.     

Proper forcing axiom and selective separability

We continue the study of Selectively Separable (SS) and, a game-theoretic strengthening, strategically selectively separable spaces (SS⁺) (see Barman, Dow (2011) [1]). The motivation for studying SS⁺ is that it is a property possessed by all separable subsets of Cp(X) for each σ-compact space X. We prove that the winning strategy for countable SS⁺ spaces can be chosen to be Markov. We introduce the notion of being compactlike for a collection of open sets in a topological space and with the help of this notion we prove that there are two countable SS⁺ spaces such that the union fails to be SS⁺, which contrasts the known result about SS spaces. We also prove that the product of two countable SS⁺ spaces is again countable SS⁺. One of the main results in this paper is that the proper forcing axiom, PFA, implies that the product of two countable Fréchet spaces is SS, a statement that was shown in Barman, Dow (2011) [1] to consistently fail. An auxiliary result is that it is consistent with the negation of CH that all separable Fréchet spaces have π-weight at most ω₁.     

Weak separation axioms and weak covering properties

We identify some remnants of normality and call them rudimentary normality, generalize the concept of submetacompact spaces to that of a weakly subparacompact space and that of a weakly^? subparacompact space, and make a simultaneous generalization of collectionwise normality and screenability with the introduction of what is to be called collectionwise σ-normality. With these weak properties, we show that,1) on weakly subparacompact spaces, countable compactness = compactness, ω₁-compactness = Lindelöfness;2) on weakly subparacompact Hausdorff spaces with rudimentary normality, regularity = normality = countable paracompactness; and3) on weakly subparacompact regular T₁-spaces with rudimentary normality, collectionwise σ-normality = screenability = collectionwise normality = paracompactness.The famous Normal Moore Space Conjecture is thus given an even more striking appearance and Worrell and Wicke?s factorization of paracompactness (over Hausdorff spaces) along with Krajewski?s are combined and strengthened. The methodology extends itself to the factorization of paracompactness on locally compact, locally connected spaces in the manner of Gruenhage and on locally compact spaces in that of Tall, and to the factorization of subparacompactness and metacompactness in the genre of Katuta, Chaber, Junnila and Price and Smith and that of Boone, improving all of them.     

Constructing aD, non-D-spaces

We introduce a general method to construct 0-dimensional, scattered T₂ spaces which are not linearly D. The construction is used to show that there are aD, non-D-spaces, answering a question of Arhangel?skii. The latter example is achieved using Shelah?s club guessing principles.     

HEREDITARY SOBRIETY AND THE CARDINALITY OF T 1 TOPOLOGIES ON COUNTABLE SETS

Abstract

A T ₀ space is called sober provided the only irreducibly closed sets are the closures of singletons; a closed set is irreducibly closed if it cannot be written as a union of two of its proper closed subsets. The relationship between hereditarily sober spaces and the lower separation axioms is examined; e.g., every hereditarily sober space satisfies axiom T _D (the derived set of every set is closed). For T ₁ spaces, hereditary sobriety is much weaker than Hausdorff, however an hereditarily sober T ₁ topology on a countably infinite set has cardinality of the continumn.     

On the tychonoff functor and w-compactness

The main purpose of this paper is to settle the following problem concerning a product formula for the Tychonoff functor τ, by introducing the notion of w-compact spaces: Characterize a topological space X such that τ(X×Y)=τ(X)×τ(Y) for any topological space Y. We also study the properties of w-compact spaces, and it is proved that, for any family {X_α} of w-compact spaces, the product ΠX_α is also w-compact and τ(ΠX_α)=Πτ(X_α).     

Remote points in large products

A point p ∈ βX\X is a remote point of X if p? cl_βXD for any nowhere dense D ? X. Van Douwen, and independently Chae and Smith, have shown that each non-pseudocompact space of countable π-weight has a remote point. Van Mill showed that many spaces of π-weight ω₁, such as ω×2^ω₁ also have remote points.We show that arbitrarily large products of spaces with countable π-weight which are not pseudocompact have remote points. In particular, ω×2^? for any infinite cardinal ?.     

On the product of homogeneous spaces

Within the class of Tychonoff spaces, and within the class of topological groups, most of the natural questions concerning ‘productive closure’ of the subclasses of countably compact and pseudocompact spaces are answered by the following three well-known results: (1) [ZFC] There is a countably compact Tychonoff space X such that X × X is not pseudocompact; (2) [ZFC] The product of any set of pseudocompact topological groups is pseudocompact; and (3) [ZFC+ MA] There are countably compact topological groups G₀, G₁ such that G₀ × G₁ is not countably compact.In this paper we consider the question of ‘productive closure” in the intermediate class of homogeneous spaces. Our principal result, whose proof leans heavily on a simple, elegant result of V.V. Uspenski?, is this: In ZFC there are pseudocompact, homogeneous spaces X₀, X₁ such that X₀ × X₁ is not pseudocompact; if in addition MA is assumed, the spaces X_i may be chosen countably compact.Our construction yields an unexpected corollary in a different direction: Every compact space embeds as a retract in a countably compact, homogeneous space. Thus for every cardinal number α there is a countably compact, homogeneous space whose Souslin number exceeds α.     

An asymmetric Ellis theorem

In 1957 Robert Ellis proved that a group with a locally compact Hausdorff topology T making all translations continuous also has jointly continuous multiplication and continuous inversion, and is thus a topological group. The theorem does not apply to locally compact asymmetric spaces such as the reals with addition and the topology of upper open rays. We first show a bitopological Ellis theorem, and then introduce a generalization of locally compact Hausdorff, called locally skew compact, and a topological dual, Tk, to obtain the following asymmetric Ellis theorem which applies to the example above:Whenever (X,⋅,T) is a group with a locally skew compact topology making all translations continuous, then multiplication is jointly continuous in both (X,⋅,T) and (X,⋅,Tk), and inversion is a homeomorphism between (X,T) and (X,Tk).This generalizes the classical Ellis theorem, because T=Tk when (X,T) is locally compact Hausdorff.