According to Markov (1946) [24], a subset of an abelian group G of the form {x ∈ G: nx = 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 Fréchet-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 Tdense 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 longstanding problem of Markov (1946) [24]. We use P and N to denote the sets of all prime numbers and all natural numbers, respectively. In this paper, 0 ∈ N. As usual, Z denotes the group of integers, and Z(n) denotes the cyclic group of order n. We use c to denote the cardinality of the continuum. The symbol ω 1 denotes the first uncountable cardinal.
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 in U_n for every n in such a way that the sequence (x_n)
has a convergent subsequence. We show that the class of SSP spaces is closed
under taking arbitrary products and continuous images, contains the class of
all dyadic spaces and forms a proper subclass of the class of strongly
pseudocompact spaces introduced recently by Garc\'ia-Ferreira and
Ortiz-Castillo. We investigate basic properties of this new class and its
relations with known compactness properties. We prove that every omega-bounded
(=the closure of which countable set is compact) group is SSP, while compact
spaces need not be SSP. Finally, we construct SSP group topologies on both the
free group and the free Abelian group with continuum-many generators
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