Nondiscrete P-groups can be reflexive

We present the first examples of nondiscrete reflexive P-groups (topological groups in which countable intersections of open sets are open) as well as of noncompact reflexive ω-bounded groups (precompact groups in which the closure of every countable set is compact). Our main result implies that every product of discrete Abelian groups equipped with the P-modified topology is reflexive. Taking uncountably many nontrivial factors, we thus answer a question posed by P. Nickolas and solve a problem raised by Ardanza-Trevijano, Chasco, Domínguez, and Tkachenko.New examples of non-reflexive P-groups are also given which are based on a further development of Leptin's technique going back to 1955.     

Characterization of almost maximally almost-periodic groups

Let G be an Abelian group. We prove that a group G admits a Hausdorff group topology τ such that the von Neumann radical n(G,τ) of (G,τ) is non-trivial and finite iff G has a non-trivial finite subgroup. If G is a topological group, then n(n(G))≠n(G) if and only if n(G) is not dually embedded. In particular, n(n(Z,τ))=n(Z,τ) for any Hausdorff group topology τ on Z.     

On extremally disconnected topological groups

We introduce the notion of a partially selective ultrafilter and prove that (a) if G is an extremally disconnected topological group and p is a converging nonprincipal ultrafilter on G containing a countable discrete subset, then p is partially selective, and (b) the existence of a nonprincipal partially selective ultrafilter on a countable set implies the existence of a P-point in ω^∗. Thus it is consistent with ZFC that there is no extremally disconnected topological group containing a countable discrete nonclosed subset.     

First countability, tightness, and other cardinal invariants in remainders of topological groups

We consider the following natural questions: when a topological group G has a first countable remainder, when G has a remainder of countable tightness? This leads to some further questions on the properties of remainders of topological groups. Let G be a topological group. The following facts are established. 1. If Gω has a first countable remainder, then either G is metrizable, or G is locally compact. 2. If G has a countable network and a first countable remainder, then either G is separable and metrizable, or G is σ-compact. 3. Under (MA+¬CH) every topological group with a countable network and a first countable remainder is separable and metrizable. Some new open problems are formulated.     

Irresolvable left topological groups

We prove that an irresolvable left topological group is of the first category. The pseudocharacter of an irresolvable left topological groupG is countable, provided thatG is Abelian or its cardinality is nonmeasurable. Some other cardinal invariants of an irresolvable left topological group are also determined. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 52, No. 6, pp. 758–765, June, 2000.     

Group duality with the topology of precompact convergence

The Pontryagin-van Kampen (P-vK) duality, defined for topological Abelian groups, is given in terms of the compact-open topology. Polar reflexive spaces, introduced by Köthe, are those locally convex spaces satisfying duality when the dual space is equipped with the precompact-open topology. It is known that the additive groups of polar reflexive spaces satisfy P-vK duality. In this note we consider the duality of topological Abelian groups when the topology of the dual is the precompact-open topology. We characterize the precompact reflexive groups, i.e., topological groups satisfying the group duality defined in terms of the precompact-open topology. As a consequence, we obtain a new characterization of polar reflexive spaces. We also present an example of a space which satisfies P-vK duality and is not polar reflexive. Some of our results respond to questions appearing in the literature.     

Proximinality in Banach spaces

In this paper, we study approximatively τ-compact and τ-strongly Chebyshev sets, where τ is the norm or the weak topology. We show that the metric projection onto τ-strongly Chebyshev sets are norm-τ continuous. We characterize approximatively τ-compact and τ-strongly Chebyshev hyperplanes and use them to characterize factor reflexive proximinal subspaces in τ-almost locally uniformly rotund spaces. We also prove some stability results on approximatively τ-compact and τ-strongly Chebyshev subspaces.     

Characterizing sequences for precompact group topologies

Motivated from [31], call a precompact group topology τ on an abelian group G ss-precompact (abbreviated from single sequence precompact  ) if there is a sequence u=(_un)$\mathbf{u}=(u_n)$ in G such that τ is the finest precompact group topology on G   making u=(_un)$\mathbf{u}=(u_n)$ converge to zero. It is proved that a metrizable precompact abelian group (G,τ)$(G,\tau )$ is ss-precompact iff it is countable. For every metrizable precompact group topology τ on a countably infinite abelian group G there exists a group topology η such that η is strictly finer than τ   and the groups (G,τ)$(G,\tau )$ and (G,η)$(G,\eta )$ have the same Pontryagin dual groups (in other words, (G,τ)$(G,\tau )$ is not a Mackey group in the class of maximally almost periodic groups).     

Embedding paratopological groups into topological products

We show that a Hausdorff paratopological group G admits a topological embedding as a subgroup into a topological product of Hausdorff first-countable (second-countable) paratopological groups if and only if G is ω-balanced (totally ω-narrow) and the Hausdorff number of G is countable, i.e., for every neighbourhood U of the neutral element e of G there exists a countable family γ of neighbourhoods of e such that _?V∈γVV−1⊆U. Similarly, we prove that a regular paratopological group G can be topologically embedded as a subgroup into a topological product of regular first-countable (second-countable) paratopological groups if and only if G is ω-balanced (totally ω-narrow) and the index of regularity of G is countable.As a by-product, we show that a regular totally ω-narrow paratopological group with countable index of regularity is Tychonoff.     

Uncountable products of determined groups need not be determined

If H is a dense subgroup of G, we say that H determines G if their groups of characters are topologically isomorphic when equipped with the compact open topology. If every dense subgroup of G determines G, then we say that G is determined. The importance of this property is justified by the recent generalizations of Pontryagin-van Kampen duality to wider classes of topological Abelian groups. Among other results, we show (a) _⊕i∈IR determines the product _∏i∈IR if and only if I is countable, (b) a compact group is determined if and only if its weight is countable. These answer questions of Comfort, Raczkowski and the third listed author. Generalizations of the above results are also given.     

Locally minimal topological groups 1

The aim of this paper is to go deeper into the study of local minimality and its connection to some naturally related properties. A Hausdorff topological group (G,τ) is called locally minimal if there exists a neighborhood U of 0 in τ such that U fails to be a neighborhood of zero in any Hausdorff group topology on G which is strictly coarser than τ. Examples of locally minimal groups are all subgroups of Banach-Lie groups, all locally compact groups and all minimal groups. Motivated by the fact that locally compact NSS groups are Lie groups, we study the connection between local minimality and the NSS property, establishing that under certain conditions, locally minimal NSS groups are metrizable. A symmetric subset of an abelian group containing zero is said to be a GTG set if it generates a group topology in an analogous way as convex and symmetric subsets are unit balls for pseudonorms on a vector space. We consider topological groups which have a neighborhood basis at zero consisting of GTG sets. Examples of these locally GTG groups are: locally pseudoconvex spaces, groups uniformly free from small subgroups (UFSS groups) and locally compact abelian groups. The precise relation between these classes of groups is obtained: a topological abelian group is UFSS if and only if it is locally minimal, locally GTG and NSS. We develop a universal construction of GTG sets in arbitrary non-discrete metric abelian groups, that generates a strictly finer non-discrete UFSS topology and we characterize the metrizable abelian groups admitting a strictly finer non-discrete UFSS group topology. Unlike the minimal topologies, the locally minimal ones are always available on “large” groups. To support this line, we prove that a bounded abelian group G admits a non-discrete locally minimal and locally GTG group topology iff |G|?c.     

Monads in topology

Let T be a submonad of the ultrafilter monad β and let G be a subfunctor of the filter functor. The T-algebras are topological spaces whose closed sets are the subalgebras and form thereby an equationally definable full subcategory of topological spaces. For appropriate T, countably generated free algebras provide ZFC examples of separable, Urysohn, countably compact, countably tight spaces which are neither compact nor sequential, and c² non-homeomorphic such examples exist. For any space X, say that U⊂X is G-open if U belongs to every ultrafilter in GX which converges in U. The full subcategory TopG consists of all G-spaces, those spaces in which every G-open set is open. Each TopG has at least these stability properties: it contains all Alexandroff spaces, and is closed under coproducts, quotients and locally closed subspaces. Examples include sequential spaces, P-spaces and countably tight spaces. T-algebras are characterized as the T-compact, T-Hausdorff T-spaces. Malyhin's theorem on countable tightness generalizes verbatim to TopG for any G⊂β. For r∈ω^?=βω\ω, let Gr be the subfunctor of β generated by r and let Tr be the generated submonad. If RK_? is the Rudin-Keisler preorder on ω^?, rRK_?s⇔Gr⊂Gs. Let c_? be the Comfort preorder and define the monadic preorderrm_?s to mean Tr⊂Ts. Then rRK_?s⇒rm_?s⇒rc_?s. It follows that there exist c² monadic types. For each such type Tr, the Tr-algebras form an equationally definable full subcategory of topological spaces with only one operation of countably infinite arity. No two of these varieties are term equivalent nor is any one a full subcategory of another inside topological spaces. Say that r∈ω^? is an m-point if Gr≠Tr. Under CH, m-points exist.     

Pseudocompact group topologies with no infinite compact subsets

We show that every Abelian group satisfying a mild cardinal inequality admits a pseudocompact group topology from which all countable subgroups inherit the maximal totally bounded topology (we say that such a topology satisfies property ).Every pseudocompact Abelian group G with cardinality |G|≤2^2c satisfies this inequality and therefore admits a pseudocompact group topology with property . Under the Singular Cardinal Hypothesis (SCH) this criterion can be combined with an analysis of the algebraic structure of pseudocompact groups to prove that every pseudocompact Abelian group admits a pseudocompact group topology with property .We also observe that pseudocompact Abelian groups with property contain no infinite compact subsets and are examples of Pontryagin reflexive precompact groups that are not compact.     

On M-separability of countable spaces and function spaces

We study M-separability as well as some other combinatorial versions of separability. In particular, we show that the set-theoretic hypothesis b=d implies that the class of selectively separable spaces is not closed under finite products, even for the spaces of continuous functions with the topology of pointwise convergence. We also show that there exists no maximal M-separable countable space in the model of Frankiewicz, Shelah, and Zbierski in which all closed P-subspaces of ω^* admit an uncountable family of nonempty open mutually disjoint subsets. This answers several questions of Bella, Bonanzinga, Matveev, and Tkachuk.     

The Euler Class in homological algebra

Let G_τ be the topological group of orientation preserving homeomorphisms of the circle, and G_δ the same group with the discrete topology. Motivated by the classical problem of reducing a circle bundle with structure group G_τ to a totally disconnected subgroup K⊂G_δ, and more currently, applications to mapping class groups, we analyze, in a homological algebra setting, the role played by the Topological and Discrete Euler Classes. In particular we describe the Discrete Euler Class of G, and any of its subgroups K, explicitly as a group extension. We apply our constructions to show that the values of the Discrete Euler Class are bounded on any space, and we state triviality and non-triviality conditions for its powers in the based mapping class groups.     

On the Ultrafilter Semigroup of an Abelian Topological Group

The ultrafilter semigroup of a topological group G, denoted Ult(G), consists of all nonprincipal ultrafilters on G converging to the identity and is a closed subsemigroup in the Stone-Cech compactification β G_d of G as a discrete semigroup. We show that for every countable nondiscrete Abelian topological group G not containing an open Boolean subgroup, the structural group of the smallest ideal of Ult(G) has cardinality 2^c.     

Local spectral synthesis on Abelian groups

We say that a complex valued function defined on an Abelian group G is a local polynomial, if its restriction to every finitely generated subgroup of G is a polynomial. We prove that local spectral synthesis (that is, spectral synthesis using local polynomials instead of polynomials) holds on every Abelian group having countable torsion free rank. More precisely, there is a cardinal ω ₁≦κ≦2 ^ω such that local spectral synthesis holds on an Abelian group G if and only if the torsion free rank of G is less than κ.     

Embeddings and frames of function spaces

For every Tychonoff space X we denote by Cp(X) the set of all continuous real-valued functions on X with the pointwise convergence topology, i.e., the topology of subspace of RX. A set P is a frame for the space Cp(X) if Cp(X)⊂P⊂RX. We prove that if Cp(X) embeds in a σ-compact space of countable tightness then X is countable. This shows that it is natural to study when Cp(X) has a frame of countable tightness with some compactness-like property. We prove, among other things, that if X is compact and the space Cp(X) has a Lindelöf frame of countable tightness then t(X)?ω. We give some generalizations of this result for the case of frames as well as for embeddings of Cp(X) in arbitrary spaces.     

Functorial topologies and finite-index subgroups of abelian groups

In the general context of functorial topologies, we prove that in the lattice of all group topologies on an abelian group, the infimum between the Bohr topology and the natural topology is the profinite topology. The profinite topology and its connection to other functorial topologies is the main objective of the paper. We are particularly interested in the poset C(G) of all finite-index subgroups of an abelian group G, since it is a local base for the profinite topology of G. We describe various features of the poset C(G) (its cardinality, its cofinality, etc.) and we characterize the abelian groups G for which C(G)?{G} is cofinal in the poset of all subgroups of G ordered by inclusion. Finally, for pairs of functorial topologies T, S we define the equalizer E(T,S), which permits to describe relevant classes of abelian groups in terms of functorial topologies.     

Pontryagin duality for topological Abelian groups

A topological Abelian group G is Pontryagin reflexive, or P-reflexive for short, if the natural homomorphism of G to its bidual group is a topological isomorphism. We look at the question, set by Kaplan in 1948, of characterizing the topological Abelian groups that are P-reflexive. Thus, we find some conditions on an arbitrary group G that are equivalent to the P-reflexivity of G and give an example that corrects a wrong statement appearing in previously existent characterizations of P-reflexive groups. Received: 10 February 2000 / Published online: 17 May 2001