S. Gabriyelyan
We prove that every reflexive abelian group $G$ such that its dual group $G^\wedge$ has the $qc$-Glicksberg property is a Mackey group. We show that a reflexive abelian group of finite exponent is a Mackey group. We prove that, for a realcompact space $X$, the space $C_k(X)$ is barrelled if and only if it is a Mackey group.
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We give the first example of a locally quasi-convex (even countable reflexive and $k_ω$) abelian group $G$ which does not admit the strongest compatible locally quasi-convex group topology. Our group $G$ is the Graev free abelian group $A_G(\mathbf{s})$ over a convergent sequence $\mathbf{s}$.
In this article it is shown that to every non-discrete Hausdorff linear topology on $\Z$ other metrizable locally quasi-convex group topologies can be associated which are strictly finer than the linear topology and such that the character groups coincide. Applying this result to the $p$-adic topology on $\Z$, we give a negative answer to the question of Dikranjan, whether this topology is Mackey.
Recently Außenhofer and the author independently have shown that the free abelian topological group $A(\mathbf{s})$ over a convergent sequence $\mathbf{s}$ does not admit the strongest compatible locally quasi-convex group topology that gives the first example of a locally quasi-convex abelian group without a Mackey group topology. In this note we considerably extend this example by showing that the free abelian topological group $A(X)$ over a non-discrete zero-dimensional metrizable space $X$ does not have a Mackey group topology. In particular, for every countable non-discrete metrizable spa…
A topological group $(G,μ)$ from a class $\mathcal G$ of MAP topological abelian groups will be called a {\it Mackey group} in $\mathcal G$ if it has the following property: if $ν$ is a group topology in $G$ such that $(G,ν)\in \mathcal G$ and $(G,ν)$ has the same continuous characters, say $(G,ν)^{\wedge}=(G,μ)^{\wedge}$, then $ν\le μ$. If $\rm{LCS}$ is the class of Hausdorff topological abelian groups which admit a structure of a locally convex topological vector space over $\mathbb R$, it is well-known that every metrizable $(G,μ) \in \rm{LCS}$ is a Mackey group in $\rm{LCS}$. For the class…
We present a series of examples of nondiscrete reflexive P-groups (i.e., groups in which all $G_δ$-sets are open) as well as noncompact reflexive $ω$-bounded groups (in which the closure of every countable set is compact). Our main result implies that every product of feathered (equivalently, almost metrizable) Abelian groups equipped with the P-modified topology is a reflexive group. In particular, every compact Abelian group with the P-modified topology is reflexive. This answers a question posed by S. Hernández and P. Nickolas and solves a problem raised by Ardanza-Trevijano, Chasco, Dom\'ı…
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.
It is proved that any infinite Abelian group of infinite exponent admits a non-discrete reflexive group topology.
For a Tychonoff space $X$, denote by $\mathfrak{P}$ the family of topological properties $\mathcal{P}$ of being a convergent sequence or being a compact, sequentially compact, countably compact, pseudocompact and functionally bounded subset of $X$, respectively. A maximally almost periodic $(MAP$) group $G$ respects $\mathcal{P}$ if $\mathcal{P}(G)=\mathcal{P}(G^+)$, where $G^+$ is the group $G$ endowed with the Bohr topology. We study relations between different respecting properties from $\mathfrak{P}$ and show that the respecting convergent sequences (=the Schur property) is the weakest one…
We prove that the group G=Hom(P,Z) of all homomorphisms from the Baer-Specker group P to the group Z of integer numbers endowed with the topology of pointwise convergence contains no infinite compact subsets. We deduce from this fact that the second Pontryagin dual of G is discrete. As G is non-discrete, it is not reflexive. Since G can be viewed as a closed subgroup of the Tychonoff product of continuum many copies of the integers Z, this provides an example of a group described in the title, thereby answering Problem 11 from [J.Galindo, L.Recorder-Núñez, M.Tkachenko, Reflexivity of prodiscre…
Recall that a space X is selectively pseudocompact if for every sequence (U_n) of non-empty open subsets of X one can choose a point x_n in U_n for all n such that the resulting sequence (x_n) has an accumulation point in X. This notion was introduced under the name strong pseudocompactness by García-Ferreira and Ortiz-Castillo, the present name is due to Dorantes-Aldama and the first author. In 2015, García-Ferreira and Tomita constructed a pseudocompact Boolean group that is not selectively pseudocompact. We prove that if the subgroup topology on every countable subgroup H of an infinite Boo…
Levine Equivalence in Aura Topological Spaces: Reachability, Alexandrov Structure, and Quotient Frames
本文研究aura拓扑空间中的Levine等价，证明其为Alexandrov拓扑，给出aura-Levine包的显式表示，刻画等价条件与商结构，对有限空间给出等价判定步骤。
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…
Tensor products of topological abelian groups and Pontryagin duality
本文解决了Baer-Specker群的Z-值同态群的对偶群公开问题，证明其拓扑同构于Z^N与T的局部拟凸张量积，还研究了两类拓扑阿贝尔群的Pontryagin自反性并得到相关结论。
This paper deals mainly with the Chu duality of discrete groups. Among other results, we give sufficient conditions for an $FC$ group to satisfy Chu duality and characterize when the Chu quasi-dual and the Takahashi quasi-dual of a group $G$ coincide. As a consequence, it follows that when $G$ is a weak sum of a family of finite simple groups, if the exponent of the groups in the family is bounded then $G$ satisfies Chu duality; on the other hand, if the exponent of the groups goes to infinite then the Chu quasi-dual of $G$ coincide with its Takahashi quasi-dual. We also present examples of di…
We provide some characterizations of precompact abelian groups $G$ whose dual group $G_p^\wedge$ endowed with the pointwise convergence topology on elements of $G$ contains a nontrivial convergent sequence. In the special case of precompact abelian \emph{torsion} groups $G$, we characterize the existence of a nontrivial convergent sequence in $G_p^\wedge$ by the following property of $G$: \emph{No infinite quotient group of $G$ is countable.} Finally, we present an example of a dense subgroup $G$ of the compact metrizable group $\mathbb{Z}(2)^ω$ such that $G$ is of the first category in itself…
Asplund spaces and the finest locally convex topology
本文在局部凸空间框架下研究Asplund性质，证明自由局部凸空间非GDS且无无限维Baire子空间，并给出quojection的Asplund等价条件。
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