The notion of locally quasi-convex abelian group, introduce by Vilenkin, is extended to maximally almost-periodic non-necessarily abelian groups. For that purpose, we look at certain bornologies that can be defined on the set $\hbox{rep}(G)$ of all finite dimensional continuous representations on a topological group $G$ in order to associate well behaved group topologies (dual topologies) to them. As a consequence, the lattice of all Hausdorff totally bounded group topologies on a group $G$ is shown to be isomorphic to the lattice of certain special subsets of $\hbox{rep}(G_d)$. Moreover, generalizing some ideas of Namioka, we relate the structural properties of the dual topological groups to topological properties of the bounded subsets belonging to the associate bornology. In like manner, certain type of bornologies that can be defined on a group $G$ allow one to define canonically associate uniformities on the dual object $\hat G$. As an application, we prove that if for every dense subgroup $H$ of a compact group $G$ we have that $\hat H$ and $\hat G$ are uniformly isomorphic, then $G$ is metrizable. Thereby, we extend to non-abelian groups some results previously considered for abelian topological groups.
In this note we show that if $G$ is a countably infinite abelian group such that $nG=0$ for some integer $n$, then the only locally minimal group topology on $G$ is the discrete one.
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…
Under $\mathfrak{p} = \mathfrak{c}$, we answer Question 24 of \cite{dikranjan&shakhmatov3} for cardinality ${\mathfrak c}$ , by showing that if a non-torsion Abelian group of size continuum admits a countably compact Hausdorff group topology, then it admits a countably compact Hausdorff group topology with non-trivial convergent sequences.
We obtain a forcing construction that shows that it is consistent that the torsion-free Abelian group $\mathbb{Q}^{(λ)}$ admits a Hausdorff group topology which is also $\mathcal{U}$-compact and contains no non-trivial convergent sequences, where $λ$ is a cardinal whose cofinality is $ω$ and $\mathcal{U}$ is a selective ultrafilter. This answers a question posed in arXiv:1904.05928.
For a locally quasi-convex (lqc) abelian group $G$, we give the first description of all compatible group topologies on $G$ and apply this result to the Mackey group problem for lqc groups. We characterize lqc abelian groups which are Mackey groups or admit a Mackey group topology and provide a characterization of two Mackey groups whose product is Mackey. We obtain the first characterization of locally convex spaces which are Mackey groups.
We prove that the existence of a selective ultrafilter implies the existence of a countably compact Hausdorff group topology on the free Abelian group of size continuum. As a consequence, we show that the existence of a selective ultrafilter implies the existence of a Wallace semigroup (i.e., a countably compact both-sided cancellative topological semigroup which is not a topological group).
We prove that if there are $\mathfrak c$ incomparable selective ultrafilters then, for every infinite cardinal $κ$ such that $κ^ω=κ$, there exists a group topology on the free Abelian group of cardinality $κ$ without nontrivial convergent sequences and such that every finite power is countably compact. In particular, there are arbitrarily large countably compact groups. This answers a 1992 question of D. Dikranjan and D. Shakhmatov.
(1) Every infinite, Abelian compact (Hausdorff) group K admits 2^|K|-many dense, non-Haar-measurable subgroups of cardinality |K|. When K is nonmetrizable, these may be chosen to be pseudocompact. (2) Every infinite Abelian group G admits a family A of 2^2^|G|-many pairwise nonhomeomorphic totally bounded group topologies such that no nontrivial sequence in G converges in any of the topologies T in A. (For some G one may arrange w(G,T) < 2^|G| for some T in A.) (3) Every infinite Abelian group $G$ admits a family B of 2^2^|G|-many pairwise nonhomeomorphic totally bounded group topologies, with…
Even in spaces of formal power series is required a topology in order to legitimate some operations, in particular to compute infinite summations. Many topologies can be exploited for different purposes. Combinatorists and algebraists may think to usual order topologies, or the product topology induced by a discrete coefficient field, or some inverse limit topologies. Analysists will take into account the valued field structure of real or complex numbers. As the main result of this paper we prove that the topological dual spaces of formal power series, relative to the class of product topologi…
A space X is selectively sequentially pseudocompact if for every sequence (U_n) of non-empty open subsets of X, one can choose a point x_n in each U_n in such a way that the sequence (x_n) has a convergent subsequence. Let G be a group from one of the following three classes: (i) V-free groups, where V is an arbitrary variety of Abelian groups; (ii) torsion Abelian groups; (iii) torsion-free Abelian groups. Under the Singular Cardinal Hypothesis SCH, we prove that if G admits a pseudocompact group topology, then it can also be equipped with a selectively sequentially pseudocompact group topolo…
Let G be an abelian group. For a subset A of G, Cyc(A) denotes the set of all elements x of G such that the cyclic subgroup generated by x is contained in A, and G is said to have the small subgroup generating property (abbreviated to SSGP) if the smallest subgroup of G generated by Cyc(U) is dense in G for every neighbourhood U of zero of G. SSGP groups form a proper subclass of the class of minimally almost periodic groups. Comfort and Gould asked for a characterization of abelian groups G which admit an SSGP group topology, and they solved this problem for bounded torsion groups (which have…
The paper deals with group dualities. A group duality is simply a pair $(G, H)$ where $G$ is an abstract abelian group and $H$ a subgroup of characters defined on $G$. A group topology $τ$ defined on $G$ is {\it compatible} with the group duality (also called dual pair) $(G, H)$ if $G$ equipped with $τ$ has dual group $H$. A topological group $(G, τ)$ gives rise to the natural duality $(G, G^\wedge)$, where $G^\wedge$ stands for the group of continuous characters on $G$. We prove that the existence of a $g$-barrelled topology on $G$ compatible with the dual pair $(G, G^\wedge)$ is equivalent t…
Every proper closed subgroup of a connected Hausdorff group must have index at least c, the cardinality of the continuum. 70 years ago Markov conjectured that a group G can be equipped with a connected Hausdorff group topology provided that every subgroup of G which is closed in all Hausdorff group topologies on G has index at least c. Counter-examples in the non-abelian case were provided 25 years ago by Pestov and Remus, yet the problem whether Markov's Conjecture holds for abelian groups G remained open. We resolve this problem in the positive.
We study final group topologies and their relations to compactness properties. In particular, we are interested in situations where a colimit or direct limit is locally compact, a k_ω-space, or locally k_ω. As a first application, we show that unitary forms of complex Kac-Moody groups can be described as the colimit of an amalgam of subgroups (in the category of Hausdorff topological groups, and the category of k_ω-groups). Our second application concerns Pontryagin duality theory for the classes of almost metrizable topological abelian groups, resp., locally k_ωtopological abelian groups, whi…
In this paper we give two families of non-metrizable topologies on the group of the integers having a countable dual group which is isomorphic to a infinite torsion subgroup of the unit circle in the complex plane. Both families are related to $D$-sequences, which are sequences of natural numbers such that each term divides the following. The first family consists of locally quasi-convex group topologies. The second consists of complete topologies which are not locally quasi-convex. In order to study the dual groups for both families we need to make numerical considerations of independent inte…
We prove that an abelian group admits a minimally almost periodic (MinAP) group topology if and only if it is connected in its Markov-Zariski topology. In particular, every unbounded abelian group admits a MinAP group topology. This answers positively a question set by Comfort, as well as several weaker forms proposed recently by Gabriyelyan. Using this characterization we answer also two open questions of Gould. We prove that a subgroup H of an abelian group G can be realized as the von Neumann kernel of G equipped with some Hausdorff group topology if and only if H is contained in the connec…
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 $\h$). Every pseudocompact Abelian group $G$ with cardinality $|G|\leq 2^{2^\cc}$ satisfies this inequality and therefore admits a pseudocompact group topology with property $\h$. 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 Abeli…
A Hausdorff topological group G is minimal if every continuous isomorphism f: G --> H between G and a Hausdorff topological group H is open. Significantly strengthening a 1981 result of Stoyanov, we prove the following theorem: For every infinite minimal abelian group G there exists a sequence {σ_n : n\in N} of cardinals such that w(G) = sup {σ_n : n \in N} and sup {2^{σ_n} : n \in N} \leq |G| \leq 2^{w(G)}, where w(G) is the weight of G. If G is an infinite minimal abelian group, then either |G| = 2^σfor some cardinal σ, or w(G) = min {σ: |G| \leq 2^σ}; moreover, the equality |G| = 2^{w(G)} h…
In this paper we introduce the concept of MV-topology, a special class of fuzzy topological spaces, and prove a proper extension of Stone Duality to the categories of limit cut complete MV-algebras and Stone MV-spaces, namely, zero-dimensional compact Hausdorff MV-topological spaces. Then we describe the object class of limit cut complete MV-algebras, and show that any semisimple MV-algebra has a limit cut completion, namely, a minimum limit cut complete extension. Last, we compose our duality with other known ones, thus obtaining new categorical equivalences and dualities involving categories…
Problem 540 of J. D. Lawson and M. Mislove in Open Problems in Topology asks whether the process of taking duals terminate after finitely many steps with topologies that are duals of each other. The problem for $T_1$ spaces was already solved by G. E. Strecker in 1966. For certain topologies on hyperspaces (which are not necessarily $T_1$), the main question was in the positive answered by Bruce S. Burdick and his solution was presented on The First Turkish International Conference on Topology in Istanbul in 2000. In this paper we bring a complete and positive solution of the problem for all t…
We equip the product of countably many copies of a compact Abelian group X with the uniform topology, and study some properties of the topological group G thus obtained. In particular, we determine the cardinality of the dual group of G, when X is the circle group: it is precisely 2^c.
A sequence $\{a_n\}$ in a group $G$ is a {\em $T$-sequence} if there is a Hausdorff group topology $τ$ on $G$ such that $a_n\stackrelτ\longrightarrow 0$. In this paper, we provide several sufficient conditions for a sequence in an abelian group to be a $T$-sequence, and investigate special sequences in the Prüfer groups $\mathbb{Z}(p^\infty)$. We show that for $p\neq 2$, there is a Hausdorff group topology $τ$ on $\mathbb{Z}(p^\infty)$ that is determined by a $T$-sequence, which is close to being maximally almost-periodic--in other words, the von Neumann radical $\mathbf{n}(\mathbb{Z}(p^\infty…
We discuss the class of paratopological groups which admits a transversal, $T_{1}$-independent and $T_{1}$-complementary paratopological group topology. We show that the Sorgenfrey line does not admit a $T_{1}$-complementary Hausdorff paratopological group topology, which gives a negative answer to \cite[Problem 10]{AT2017}. We give a very useful criterion for transversality in term of submaximal paratopological group topology, and prove that if a non-discrete paratopological group topology $G$ contains a central subgroup which admits a transversal paratopological group topology, then so does …
For a topological monoid S the dual inverse monoid is the topological monoid of all identity preserving homomorphisms from S to the circle with attached zero. A topological monoid S is defined to be reflexive if the canonical homomorphism from S to its second dual inverse monoid is a topological isomorphism. We prove that a (compact or discrete) topological inverse monoid S is reflexive (if and) only if S is abelian and the idempotent semilattice of S is zero-dimensional. For a discrete (resp. compact) topological monoid its dual inverse monoid is compact (resp. discrete). These results unify …
