The three space problem in topological groups

We study compact, countably compact, pseudocompact, and functionally bounded sets in extensions of topological groups. A property P is said to be a three space property if, for every topological group G and a closed invariant subgroup N of G, the fact that both groups N and G/N have P implies that G also has P. It is shown that if all compact (countably compact) subsets of the groups N and G/N are metrizable, then G has the same property. However, the result cannot be extended to pseudocompact subsets, a counterexample exists under p=c. Another example shows that extensions of groups do not preserve the classes of realcompact, Dieudonné complete and μ-spaces: one can find a pseudocompact, non-compact Abelian topological group G and an infinite, closed, realcompact subgroup N of G such that G/N is compact and all functionally bounded subsets of N are finite. Several examples given in the article destroy a number of tempting conjectures about extensions of topological groups.     

Spaces of maps into topological group with the Whitney topology

Let X be a locally compact Polish space and G a non-discrete Polish ANR group. By C(X,G), we denote the topological group of all continuous maps endowed with the Whitney (graph) topology and by Cc(X,G) the subgroup consisting of all maps with compact support. It is known that if X is compact and non-discrete then the space C(X,G) is an l₂-manifold. In this article we show that if X is non-compact and not end-discrete then Cc(X,G) is an (R^∞×l₂)-manifold, and moreover the pair (C(X,G),Cc(X,G)) is locally homeomorphic to the pair of the box and the small box powers of l₂.     

Projective versions of selection principles

All spaces are assumed to be Tychonoff. A space X is called projectively P (where P is a topological property) if every continuous second countable image of X is P. Characterizations of projectively Menger spaces X in terms of continuous mappings , of Menger base property with respect to separable pseudometrics and a selection principle restricted to countable covers by cozero sets are given. If all finite powers of X are projectively Menger, then all countable subspaces of Cp(X) have countable fan tightness. The class of projectively Menger spaces contains all Menger spaces as well as all σ-pseudocompact spaces, and all spaces of cardinality less than d. Projective versions of Hurewicz, Rothberger and other selection principles satisfy properties similar to the properties of projectively Menger spaces, as well as some specific properties. Thus, X is projectively Hurewicz iff Cp(X) has the Monotonic Sequence Selection Property in the sense of Scheepers; βX is Rothberger iff X is pseudocompact and projectively Rothberger. Embeddability of the countable fan space Vω into Cp(X) or Cp(X,2) is characterized in terms of projective properties of X.     

On nearly pseudocompact spaces

A completely regular space X is called nearly pseudocompact if υX?X is dense in βX?X, where βX is the Stone-?ech compactification of X and υX is its Hewitt realcompactification. After characterizing nearly pseudocompact spaces in a variety of ways, we show that X is nearly pseudocompact if it has a dense locally compact pseudocompact subspace, or if no point of X has a closed realcompact neighborhood. Moreover, every nearly pseudocompact space X is the union of two regular closed subsets X₁, X₂ such that Int X₁ is locally compact, no points of X₂ has a closed realcompact neighborhood, and $\text{Int(X}{}_1\text{?X}{}_2\text{)=?}$. It follows that a product of two nearly pseudocompact spaces, one of which is locally compact, is also nearly pseudocompact.     

A note on D-spaces

We introduce notions of nearly good relations and N-sticky modulo a relation as tools for proving that spaces are D-spaces. As a corollary to general results about such relations, we show that Cp(X) is hereditarily a D-space whenever X is a Lindelöf Σ-space. This answers a question of Matveev, and improves a result of Buzyakova, who proved the same result for X compact.We also prove that if a space X is the union of finitely many D-spaces, and has countable extent, then X is linearly Lindelöf. It follows that if X is in addition countably compact, then X must be compact. We also show that Corson compact spaces are hereditarily D-spaces. These last two results answer recent questions of Arhangel'skii. Finally, we answer a question of van Douwen by showing that a perfectly normal collectionwise-normal non-paracompact space constructed by R. Pol is a D-space.     

G-compactifications of pseudocompact G-spaces

Examples of a pseudocompact (even countably compact) G-space which is not G-Tychonoff and of a locally compact pseudocompact (even countably compact) G-Tychonoff space X with βGX≠βX are constructed.     

Group-valued continuous functions with the topology of pointwise convergence

Let G be a topological group with the identity element e. Given a space X, we denote by Cp(X,G) the group of all continuous functions from X to G endowed with the topology of pointwise convergence, and we say that X is: (a) G-regular if, for each closed set F⊆X and every point x∈X?F, there exist f∈Cp(X,G) and g∈G?{e} such that f(x)=g and f(F)⊆{e}; (b) G^?-regular provided that there exists g∈G?{e} such that, for each closed set F⊆X and every point x∈X?F, one can find f∈Cp(X,G) with f(x)=g and f(F)⊆{e}. Spaces X and Y are G-equivalent provided that the topological groups Cp(X,G) and Cp(Y,G) are topologically isomorphic.We investigate which topological properties are preserved by G-equivalence, with a special emphasis being placed on characterizing topological properties of X in terms of those of Cp(X,G). Since R-equivalence coincides with l-equivalence, this line of research “includes” major topics of the classical Cp-theory of Arhangel'ski? as a particular case (when G=R).We introduce a new class of TAP groups that contains all groups having no small subgroups (NSS groups). We prove that: (i) for a given NSS group G, a G-regular space X is pseudocompact if and only if Cp(X,G) is TAP, and (ii) for a metrizable NSS group G, a G^?-regular space X is compact if and only if Cp(X,G) is a TAP group of countable tightness. In particular, a Tychonoff space X is pseudocompact (compact) if and only if Cp(X,R) is a TAP group (of countable tightness). Demonstrating the limits of the result in (i), we give an example of a precompact TAP group G and a G-regular countably compact space X such that Cp(X,G) is not TAP.We show that Tychonoff spaces X and Y are T-equivalent if and only if their free precompact Abelian groups are topologically isomorphic, where T stays for the quotient group R/Z. As a corollary, we obtain that T-equivalence implies G-equivalence for every Abelian precompact group G. We establish that T-equivalence preserves the following topological properties: compactness, pseudocompactness, σ-compactness, the property of being a Lindelöf Σ-space, the property of being a compact metrizable space, the (finite) number of connected components, connectedness, total disconnectedness. An example of R-equivalent (that is, l-equivalent) spaces that are not T-equivalent is constructed.     

On one-point metrizable extensions of locally compact metrizable spaces

For a non-compact metrizable space X, let E(X) be the set of all one-point metrizable extensions of X, and when X is locally compact, let EK(X) denote the set of all locally compact elements of E(X) and be the order-anti-isomorphism (onto its image) defined in [M. Henriksen, L. Janos, R.G. Woods, Properties of one-point completions of a non-compact metrizable space, Comment. Math. Univ. Carolin. 46 (2005) 105-123; in short HJW]. By definition λ(Y)=_?n<ωclβX(Un∩X)\X, where Y=X∪{p}∈E(X) and {Un}_n<ω is an open base at p in Y. We characterize the elements of the image of λ as exactly those non-empty zero-sets of βX which miss X, and the elements of the image of EK(X) under λ, as those which are moreover clopen in βX\X. This answers a question of [HJW]. We then study the relation between E(X) and EK(X) and their order structures, and introduce a subset ES(X) of E(X). We conclude with some theorems on the cardinality of the sets E(X) and EK(X), and some open questions.     

QN-spaces, wQN-spaces and covering properties

The main results of the paper are as follows: covering characterizations of wQN-spaces, covering characterizations of QN-spaces and a theorem saying that Cp(X) has the Arkhangel'ski?ˇ property (α₁) provided that X is a QN-space. The latter statement solves a problem posed by M. Scheepers [M. Scheepers, Cp(X) and Arhangel'ski?ˇ's αi-spaces, Topology Appl. 89 (1998) 265-275] and for Tychonoff spaces was independently proved by M. Sakai [M. Sakai, The sequence selection properties of Cp(X), Preprint, April 25, 2006]. As the most interesting result we consider the equivalence that a normal topological space X is a wQN-space if and only if X has the property S₁(Γshr,Γ). Moreover we show that X is a QN-space if and only if Cp(X) has the property (α₀), and for perfectly normal spaces, if and only if X has the covering property (β₃).     

Various types of local connectedness in n-fold hyperspaces

Let X be a continuum. The n-fold hyperspace Cn(X), n<∞, is the space of all nonempty compact subsets of X with the Hausdorff metric. Four types of local connectivity at points of Cn(X) are investigated: connected im kleinen, locally connected, arcwise connected im kleinen and locally arcwise connected. Characterizations, as well as necessary or sufficient conditions, are obtained for Cn(X) to have one or another of the local connectivity properties at a given point. Several results involve the property of Kelley or C^*-smoothness. Some new results are obtained for C(X), the space of subcontinua of X. A class of continua X is given for which Cn(X) is connected im kleinen only at subcontinua of X and for which any two such subcontinua must intersect.     

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.     

Extensions and new observations of Tychonoff,Stone-Weierstrass theorems,compactifications and the realcompactification

An extension of the Tychonoff theorem is obtained in characterizing a compact space by the nets and the images induced by any family of continuous functions on it. The idea of this extension is applied to get a new process and new observations of compactifications and the realcompactification. Finally, a sufficient and necessary condition of a vector sublattice or a subalgebra of C¹(X) to be dense in (C¹(X),∥·∥) is provided in terms of the nets in X induced by C¹(X), where C¹(X) is the space of all bounded real continuous functions on a topological space X with pointwise ordering, and ∥·∥ is the supremum norm.     

Classes defined by stars and neighbourhood assignments

We apply and develop an idea of E. van Douwen used to define D-spaces. Given a topological property P, the class P^∗ dual to P (with respect to neighbourhood assignments) consists of spaces X such that for any neighbourhood assignment there is Y⊂X with Y∈P and . We prove that the classes of compact, countably compact and pseudocompact are self-dual with respect to neighbourhood assignments. It is also established that all spaces dual to hereditarily Lindelöf spaces are Lindelöf. In the second part of this paper we study some non-trivial classes of pseudocompact spaces defined in an analogous way using stars of open covers instead of neighbourhood assignments.     

A wage-type example with a pseudocompact factor

The following example is constructed without any set-theoretic assumptions beyond ZFC: There exist a hereditarily separable hereditarily Lindelöf space X and a first-countable locally compact separable pseudocompact space Y such that dim X = dimY = 0, while dim(X × Y)>0.     

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.     

Self-duality in the class of precompact groups

A topological Abelian group G is called (strongly) self-dual if there exists a topological isomorphism Φ:G→G^∧ of G onto the dual group G^∧ (such that Φ(x)(y)=Φ(y)(x) for all x,y∈G). We prove that every countably compact self-dual Abelian group is finite. It turns out, however, that for every infinite cardinal κ with κω=κ, there exists a pseudocompact, non-compact, strongly self-dual Boolean group of cardinality κ.     

Pseudocompactness of hyperspaces

We consider the following question of Ginsburg: Is there any relationship between the pseudocompactness ofXωand that of the hyperspaceX²? We do that first in the context of Mrówka-Isbell spaces Ψ(A) associated with a maximal almost disjoint (MAD) family A on ω answering a question of J. Cao and T. Nogura. The space Ψω(A) is pseudocompact for every MAD family A. We show that

(1)

(p=c) ²Ψ(A) is pseudocompact for every MAD family A.

(2)

(h<c) There is a MAD family A such that ²Ψ(A) is not pseudocompact.

We also construct a ZFC example of a space X such that Xω is pseudocompact, yet X² is not.     

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.     

The compact-open topology: A new perspective

This paper studies the compact-open topology on the set KC(X) of all real-valued functions defined on a Tychonoff space, which are continuous on compact subsets of X. In addition to metrizability, separability and second countability of this topology on KC(X), various kinds of topological properties of this topology are studied in detail. Actually the motivation for studying the compact-open topology on KC(X) lies in the attempt of having a simpler proof for the characterization of a completeness property of the compact-open topology on C(X), the set of all real-valued continuous functions on X.     

Automatic continuity of biseparating homomorphisms defined between groups of continuous functions

Let C(X,T) be the group of continuous functions of a compact Hausdorff space X to the unit circle of the complex plane T with the pointwise multiplication as the composition law. We investigate how the structure of C(X,T) determines the topology of X. In particular, which group isomorphisms H between the groups C(X,T) and C(Y,T) imply the existence of a continuous map h of Y into X such that H is canonically represented by h. Among other results, it is proved that C(X,T) determines X module a biseparating group isomorphism and, when X is first countable, the automatic continuity and representation as Banach-Stone maps for biseparating group isomorphisms is also obtained.