Dmitri Shakhmatov

The Markov–Zariski topology of an abelian group

Abstract According to Markov (1946) [24] , a subset of an abelian group G of the form { x ∈ G : n x = a } , for some integer n and some element a ∈ G , is an elementary algebraic set; finite unions of elementary algebraic sets are called algebraic sets. We prove that a subset of an abelian group G is algebraic if and only if it is closed in every precompact (= totally bounded) Hausdorff group topology on G. The family of all algebraic sets of an abelian group G forms the family of closed subsets of a unique Noetherian T 1 topology Z G on G called the Zariski, or verbal, topology of G; see Bryant (1977) [3] . We investigate the properties of this topology. In particular, we show that the Zariski topology is always hereditarily separable and Frechet–Urysohn. For a countable family F of subsets of an abelian group G of cardinality at most the continuum, we construct a precompact metric group topology T on G such that the T -closure of each member of F coincides with its Z G -closure. As an application, we provide a characterization of the subsets of G that are T -dense in some Hausdorff group topology T on G, and we show that such a topology, if it exists, can always be chosen so that it is precompact and metric. This provides a partial answer to a long-standing problem of Markov (1946) [24] .

Reflection principle characterizing groups in which unconditionally closed sets are algebraic

Abstract We give a necessary and sufficient condition, in terms of a certain reflection principle, for every unconditionally closed subset of a group G to be algebraic. As a corollary, we prove that this is always the case when G is a direct product of an Abelian group with a direct product (sometimes also called a direct sum) of a family of countable groups. This is the widest class of groups known to date where the answer to the 63-year-old problem of Markov turns out to be positive. We also prove that whether every unconditionally closed subset of G is algebraic or not is completely determined by countable subgroups of G. Essential connections with non-topologizable groups are highlighted.

Group-valued continuous functions with the topology of pointwise convergence

Abstract Let G be a topological group with the identity element e . Given a space X , we denote by C p ( 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 ∈ C p ( 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 ∈ C p ( X , G ) with f ( x ) = g and f ( F ) ⊆ { e } . Spaces X and Y are G-equivalent provided that the topological groups C p ( X , G ) and C p ( 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 C p ( X , G ) . Since R -equivalence coincides with l -equivalence, this line of research “includes” major topics of the classical C p -theory of Arhangelskiĭ 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 C p ( X , G ) is TAP, and (ii) for a metrizable NSS group G , a G ⋆ -regular space X is compact if and only if C p ( X , G ) is a TAP group of countable tightness. In particular, a Tychonoff space X is pseudocompact (compact) if and only if C p ( 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 C p ( 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 Lindelof 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.

Quasi-convex density and determining subgroups of compact abelian groups

Abstract For an abelian topological group G , let G ˆ denote the dual group of all continuous characters endowed with the compact open topology. Given a closed subset X of an infinite compact abelian group G such that w ( X ) w ( G ) , and an open neighborhood U of 0 in T , we show that | { χ ∈ G ˆ : χ ( X ) ⊆ U } | = | G ˆ | . (Here, w ( G ) denotes the weight of G .) A subgroup D of G determines G if the map r : G ˆ → D ˆ defined by r ( χ ) = χ ↾ D for χ ∈ G ˆ , is an isomorphism between G ˆ and D ˆ . We prove that w ( G ) = min { | D | : D is a subgroup of G that determines G } for every infinite compact abelian group G . In particular, an infinite compact abelian group determined by a countable subgroup is metrizable. This gives a negative answer to a question of Comfort, Raczkowski and Trigos-Arrieta (repeated by Hernandez, Macario and Trigos-Arrieta). As an application, we furnish a short elementary proof of the result from [S. Hernandez, S. Macario, F.J. Trigos-Arrieta, Uncountable products of determined groups need not be determined, J. Math. Anal. Appl. 348 (2008) 834–842] that a compact abelian group G is metrizable provided that every dense subgroup of G determines G .

Hewitt-Marczewski-Pondiczery type theorem for abelian groups and Markov's potential density

For an uncountable cardinal T and a subset S of an abelian group G, the following conditions are equivalent: (i) |{ns: s ∈ S}| ≥ T for all integers n ≥ 1; (ii) there exists a group homomorphism π: G → T 2T such that π(S) is dense in T 2t . Moreover, if |G| 1. This partially resolves a question of Markov going back to 1946.

A Kronecker–Weyl theorem for subsets of abelian groups

Abstract Let N be the set of non-negative integer numbers, T the circle group and c the cardinality of the continuum. Given an abelian group G of size at most 2 c and a countable family E of infinite subsets of G, we construct “Baire many” monomorphisms π : G → T c such that π ( E ) is dense in { y ∈ T c : n y = 0 } whenever n ∈ N , E ∈ E , n E = { 0 } and { x ∈ E : m x = g } is finite for all g ∈ G and m ∈ N ∖ { 0 } such that n = m k for some k ∈ N ∖ { 1 } . We apply this result to obtain an algebraic description of countable potentially dense subsets of abelian groups, thereby making a significant progress towards a solution of a problem of Markov going back to 1944. A particular case of our result yields a positive answer to a problem of Tkachenko and Yaschenko (2002) [22, Problem 6.5] . Applications to group actions and discrete flows on T c , Diophantine approximation, Bohr topologies and Bohr compactifications are also provided.

Productivity of sequences with respect to a given weight function

Abstract Given a function f : N → ( ω + 1 ) ∖ { 0 } , we say that a faithfully indexed sequence { a n : n ∈ N } of elements of a topological group G is: (i) f-Cauchy productive (f-productive) provided that the sequence { ∏ n = 0 m a n z ( n ) : m ∈ N } is left Cauchy (converges to some element of G, respectively) for each function z : N → Z such that | z ( n ) | ⩽ f ( n ) for every n ∈ N ; (ii) unconditionally f-Cauchy productive (unconditionally f-productive) provided that the sequence { a φ ( n ) : n ∈ N } is ( f ∘ φ ) -Cauchy productive (respectively, ( f ∘ φ ) -productive) for every bijection φ : N → N . (Bijections can be replaced by injections here.) We consider the question of existence of (unconditionally) f-productive sequences for a given “weight function” f. We prove that: (1) a Hausdorff group having an f-productive sequence for some f contains a homeomorphic copy of the Cantor set; (2) if a non-discrete group is either locally compact Hausdorff or Weil complete metric, then it contains an unconditionally f-productive sequence for every function f : N → N ∖ { 0 } ; (3) a metric group is NSS if and only if it does not contain an f ω -Cauchy productive sequence, where f ω is the function taking the constant value ω. We give an example of an f ω -productive sequence { a n : n ∈ N } in a (necessarily non-abelian) separable metric group H with a linear topology and a bijection φ : N → N such that the sequence { ∏ n = 0 m a φ ( n ) : m ∈ N } diverges, thereby answering a question of Dominguez and Tarieladze. Furthermore, we show that H has no unconditionally f ω -productive sequences. As an application of our results, we resolve negatively a question from C p ( − , G ) -theory.

On the existence of kings in continuous tournaments

Abstract 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 players is a compact Hausdorff space. The following partial converse is proved as well. Let X be a Tychonoff space which is either zero-dimensional or locally connected or pseudocompact or linearly ordered. If X admits at least one continuous tournament and each continuous tournament on X has a king, then X must be compact. We show that a complete reversal of our theorem is impossible, by giving an example of a dense connected subspace Y of the unit square admitting precisely two continuous tournaments both of which have a king, yet Y is not even analytic (much less compact).

Building suitable sets for locally compact groups by means of continuous selections

If a discrete subset S of a topological group G with the identity 1 generates a dense subgroup of G and S∪{1} is closed in G, then S is called a suitable set for G. We apply Michaels selection theorem to offer a direct, self-contained, purely topological proof of the result of Hofmann and Morris [K.-H. Hofmann, S.A. Morris, Weight and c, J. Pure Appl. Algebra 68 (1–2) (1990) 181–194] on the existence of suitable sets in locally compact groups. Our approach uses only elementary facts from (topological) group theory.

Minimal pseudocompact group topologies on free abelian groups

Abstract 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 ∈ N } of cardinals such that w ( G ) = sup { σ n : n ∈ N } and sup { 2 σ n : n ∈ N } ⩽ | G | ⩽ 2 w ( G ) , where w ( G ) is the weight of G . If G is an infinite minimal abelian group, then either | G | = 2 σ for some cardinal σ , or w ( G ) = min { σ : | G | ⩽ 2 σ } ; moreover, the equality | G | = 2 w ( G ) holds whenever cf ( w ( G ) ) > ω . For a cardinal κ , we denote by F κ the free abelian group with κ many generators. If F κ admits a pseudocompact group topology, then κ ⩾ c , where c is the cardinality of the continuum. We show that the existence of a minimal pseudocompact group topology on F c is equivalent to the Lusins Hypothesis 2 ω 1 = c . For κ > c , we prove that F κ admits a (zero-dimensional) minimal pseudocompact group topology if and only if F κ has both a minimal group topology and a pseudocompact group topology. If κ > c , then F κ admits a connected minimal pseudocompact group topology of weight σ if and only if κ = 2 σ . Finally, we establish that no infinite torsion-free abelian group can be equipped with a locally connected minimal group topology.