44 citations · 271 across the 30 of their papers we have counts for
28 papers · 1 filter
Cardinal invariants and convergence properties of locally minimal groups
Dikran Dikranjan, Dmitri Shakhmatov
If G is a locally essential subgroup of a compact abelian group K, then: (i) t(G)=w(G)=w(K), where t(G) is the tightness of G; (ii) if G is radial, then K must be metrizable; (iii)…
Compactness properties defined by open-point games
Alejandro Dorantes-Aldama, Dmitri Shakhmatov
Let S be a topological property of sequences (such as, for example, "to contain a convergent subsequence" or "to have an accumulation point"). We introduce the following open-point…
Countably compact groups and sequential order
Dmitri Shakhmatov, Alexander Shibakov
We use to construct, for every a sequential countably compact topological group of sequential order . This establishes the independence of the existenc…
SSGP topologies on free groups of infinite rank
Dmitri Shakhmatov, Víctor Hugo Yañez
We prove that every free group G with infinitely many generators admits a Hausdorff group topology T with the following property: for every T-open neighbourhood U of the identity o…
The impact of the Bohr topology on selective pseudocompactness
Dmitri Shakhmatov, Víctor Hugo Yañez
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 se…
Selectively sequentially pseudocompact group topologies on torsion and torsion-free Abelian groups
Alejandro Dorantes-Aldama, Dmitri Shakhmatov
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 sequen…
Characterizing Lie groups by controlling their zero-dimensional subgroups
Dikran Dikranjan, Dmitri Shakhmatov
We provide characterizations of Lie groups as compact-like groups in which all closed zero-dimensional metric (compact) subgroups are discrete. The "compact-like" properties we con…
Selectively pseudocompact groups without non-trivial convergent sequences
Dmitri Shakhmatov, Víctor Hugo Yañez
The existence of a countably compact group without non-trivial convergent sequences in ZFC alone is a major open problem in topological group theory. We give a ZFC example of a Boo…
SSGP topologies on abelian groups of positive finite divisible rank
Dmitri Shakhmatov, Víctor Hugo Yañez
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…
Selective sequential pseudocompactness
Alejandro Dorantes-Aldama, Dmitri Shakhmatov
We say that a topological space X is selectively sequentially pseudocompact (SSP for short) if for every sequence (U_n) of non-empty open subsets of X, one can choose a point x_n i…
The existence of continuous weak selections and orderability-type properties in products and filter spaces
Koichi Motooka, Dmitri Shakhmatov, Takamitsu Yamauchi
Orderability, weak orderability and the existence of continuous weak selections on filter spaces (i.e., spaces with a single non-isolated point) and their products are discussed. W…
Completeness and compactness properties in metric spaces, topological groups and function spaces
Alejandro Dorantes-Aldama, Dmitri Shakhmatov
We prove that many completeness properties coincide in metric spaces, precompact groups and dense subgroups of products of separable metric groups. We apply these results to functi…
A countable free closed non-reflexive subgroup of Z^c
Maria Vincenta Ferrer, Salvador Hernández, Dmitri Shakhmatov
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…
A complete solution of Markov's problem on connected group topologies
Dikran Dikranjan, Dmitri Shakhmatov
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 equi…
Direct sums and products in topological groups and vector spaces
Dikran Dikranjan, Dmitri Shakhmatov, Jan Spěvák
We call a subset of an abelian topological group : (i) provided that for every open neighbourhood of one can find a finite set $F\su…
Final solution of Protasov-Comfort's problem on minimally almost periodic group topologies
Dikran Dikranjan, Dmitri Shakhmatov
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 unbound…
Subgroups of direct products closely approximated by direct sums
Maria V. Ferrer, Salvador Hernandez, Dmitri Shakhmatov
Let be an infinite set, be a family of (topological) groups and be its direct product. For , $p_{J}: G\to \prod_{j\in J} G…
On the existence of kings in continuous tournaments
Masato Nagao, Dmitri Shakhmatov
The classical result of Landau on the existence of kings in finite tournaments (=finite directed complete graphs) is extended to continuous tournaments for which the set X of playe…
Metrization criteria for compact groups in terms of their dense subgroups
Dikran Dikranjan, Dmitri Shakhmatov
According to Comfort, Raczkowski and Trigos-Arrieta, a dense subgroup D of a compact abelian group G determines G if the restriction homomorphism G^ --> D^ of the dual groups is a…
Productivity of sequences with respect to a given weight function
Dikran Dikranjan, Dmitri Shakhmatov, Jan Spěvák
Given a function f: N --> (omega+1)-{0}, we say that a faithfully indexed sequence {a_n: n in N} of elements of a topological group G is: (i) f-Cauchy productive (f-productive) pro…
Hewitt-Marczewski-Pondiczery type theorem for abelian groups and Markov's potential density
Dikran Dikranjan, Dmitri Shakhmatov
For an uncountable cardinal τand a subset S of an abelian group G, the following conditions are equivalent: (i) |{ns:s\in S}|\ge τfor all integers n\ge 1; (ii) there exists a group…
NSS and TAP properties in topological groups close to being compact
Dikran Dikranjan, Dmitri Shakhmatov, Jan Spěvák
We introduce a notion of productivity (summability) of sequences in a topological group G, parametrized by a given function f : N --> omega+1. The extreme case when f is the functi…
Group-valued continuous functions with the topology of pointwise convergence
Dmitri Shakhmatov, Jan Spěvák
We denote by C_p(X,G) the group of all continuous functions from a space X to a topological group G endowed with the topology of pointwise convergence. We say that spaces X and Y a…
Convergent sequences in minimal groups
Dmitri Shakhmatov
A Hausdorff topological group G is minimal if every continuous isomorphism f : G --> H between G and a Hausdorff topological group H is open. Clearly, every compact Hausdorff group…