63 research outputs found
The Markov-Zariski topology of an abelian group
According to Markov, a subset of an abelian group G of the form {x in G:
nx=a}, for some integer n and some element a of 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 subsets of an abelian group G forms the family of
closed subsets of a unique Noetherian T_1 topology on G called the Zariski, or
verbal, topology of G. 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
Zariski closure. As an application, we provide a characterization of the
subsets of G that are dense in some Hausdorff group topology 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
Metrization criteria for compact groups in terms of their dense subgroups
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 topological isomorphism. We introduce four conditions
on D that are necessary for it to determine G and we resolve the following
question: If one of these conditions holds for every dense (or G_\delta-dense)
subgroup D of G, must G be metrizable? In particular, we prove (in ZFC) that a
compact abelian group determined by all its G_\delta-dense subgroups is
metrizable, thereby resolving a question by Hernandez, Macario and
Trigos-Arrieta. (Under the additional assumption of the Continuum Hypothesis
CH, the same statement was proved recently by Bruguera, Chasco, Dominguez,
Tkachenko and Trigos-Arrieta.) As a tool, we develop a machinery for building
G_\delta-dense subgroups without uncountable compact subsets in compact groups
of weight \omega_1 (in ZFC). The construction is delicate, as these subgroups
must have non-trivial convergent sequences in some models of ZFC.Comment: The exposition has substantially improved. Remarks 5.6 and 5.7 are
new. Three references adde
Hewitt-Marczewski-Pondiczery type theorem for abelian groups and Markov's potential density
For an uncountable cardinal \tau and a subset S of an abelian group G, the
following conditions are equivalent: (i) |{ns:s\in S}|\ge \tau for all integers
n\ge 1; (ii) there exists a group homomorphism \pi:G\to T^{2^\tau} such that
\pi(S) is dense in T^{2^\tau}. Moreover, if |G|\le 2^{2^\tau}, then the
following item can be added to this list: (iii) there exists an isomorphism
\pi:G\to G' between G and a subgroup G' of T^{2^\tau} such that \pi(S) is dense
in T^{2^\tau}. We prove that the following conditions are equivalent for an
uncountable subset S of an abelian group G that is either (almost) torsion-free
or divisible: (a) S is T-dense in G for some Hausdorff group topology T on G;
(b) S is T-dense in some precompact Hausdorff group topology T on G; (c)
|{ns:s\in S}|\ge \min{\tau:|G|\le 2^{2^\tau}} for every integer n\ge 1. This
partially resolves a question of Markov going back to 1946
A complete solution of Markov's problem on connected group topologies
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
Direct sums and products in topological groups and vector spaces
We call a subset of an abelian topological group : (i)
provided that for every open neighbourhood of one
can find a finite set such that the subgroup generated by
is contained in ; (ii) if, for every
family of integer numbers, there exists such that the
net \left\{\sum_{a\in F} z_a a: F\subseteq A\mbox{ is finite}\right\}
converges to ; (iii) provided that and for every neighbourhood of there exists a neighbourhood of
such that, for every finite set and each set of integers, implies that for all
. We prove that: (1) an abelian topological group contains a direct
product (direct sum) of -many non-trivial topological groups if and
only if it contains a topologically independent, absolutely (Cauchy) summable
subset of cardinality ; (2) a topological vector space contains
as its subspace if and only if it has an infinite
absolutely Cauchy summable set; (3) a topological vector space contains
as its subspace if and only if it has an
multiplier convergent series of non-zero elements.
We answer a question of Hu\v{s}ek and generalize results by
Bessaga-Pelczynski-Rolewicz, Dominguez-Tarieladze and Lipecki
Minimal pseudocompact group topologies on free abelian groups
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 {\sigma_n : n\in
N} of cardinals such that w(G) = sup {\sigma_n : n \in N} and sup {2^{\sigma_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^\sigma for some cardinal
\sigma, or w(G) = min {\sigma: |G| \leq 2^\sigma}; moreover, the equality |G| =
2^{w(G)} holds whenever cf (w(G)) > \omega. For a cardinal \kappa, we denote by
F_\kappa the free abelian group with \kappa many generators. If F_\kappa admits
a pseudocompact group topology, then \kappa \geq 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 Lusin's Hypothesis 2^{\omega_1} = c. For
\kappa > c, we prove that F_\kappa admits a (zero-dimensional) minimal
pseudocompact group topology if and only if F_\kappa has both a minimal group
topology and a pseudocompact group topology. If \kappa > c, then F_\kappa
admits a connected minimal pseudocompact group topology of weight \sigma if and
only if \kappa = 2^\sigma. Finally, we establish that no infinite torsion-free
abelian group can be equipped with a locally connected minimal group topology.Comment: 18 page
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