Large Sets in Boolean and Non-Boolean Groups and Topology

Right and left thick, syndetic, piecewise syndetic, and fat sets in groups are studied. The main concern is the interplay between such sets in Boolean groups. Natural topologies closely related to fat sets are also considered, which leads to interesting relations between fat sets and ultrafilters

Reflection principle characterizing groups in which unconditionally closed sets are algebraic

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 years 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.Comment: 14 page

The Doitchinov Completion of a Regular Paratopological Group

In memory of Professor D. Doitchinov ∗ This paper was written while the first author was supported by the Swiss National Science Foundation under grants 21–30585.91 and 2000-041745.94/1 and by the Spanish Ministry of Education and Sciences under DGES grant SAB94-0120. The second author was supported under DGES grant PB95-0737. During her stay at the University of Berne the third author was supported by the first author’s grant 2000-041745.94/1 from the Swiss National Science Foundation.We show that the two-sided quasi-uniformity UB of a regular paratopological group (G, ·) is quiet. The Doitchinov completion (G, UB ) of (G, UB ) can be considered a paratopological group containing G as a doubly dense subgroup whenever G is Abelian. Furthermore UB is the two-sided quasi-uniformity of (G, ·). These results generalize in an appropriate way important results about topological groups to regular paratopological groups. A counterexample dealing with the non-Abelian case is presented. Furthermore we give conditions, depending on quasi-uniform completeness properties, under which a paratopological group is a topological group

The topological fundamental group and free topological groups

The topological fundamental group $\pi_{1}^{top}$ is a homotopy invariant finer than the usual fundamental group. It assigns to each space a quasitopological group and is discrete on spaces which admit universal covers. For an arbitrary space $X$, we compute the topological fundamental group of the suspension space $\Sigma(X_+)$ and find that $\pi_{1}^{top}(\Sigma(X_+))$ either fails to be a topological group or is the free topological group on the path component space of $X$. Using this computation, we provide an abundance of counterexamples to the assertion that all topological fundamental groups are topological groups. A relation to free topological groups allows us to reduce the problem of characterizing Hausdorff spaces $X$ for which $\pi_{1}^{top}(\Sigma(X_+))$ is a Hausdorff topological group to some well known classification problems in topology.Comment: 33 page

Topologization of sets endowed with an action of a monoid

Given a set $X$ and a family $G$ of self-maps of $X$, we study the problem of the existence of a non-discrete Hausdorff topology on $X$ with respect to which all functions $f\in G$ are continuous. A topology on $X$ with this property is called a $G$-topology. The answer is given in terms of the Zariski $G$-topology $\zeta_G$ on $X$, that is, the topology generated by the subbase consisting of the sets $\{x\in X:f(x)\ne g(x)\}$ and $\{x\in X:f(x)\ne c\}$, where $f,g\in G$ and $c\in X$. We prove that, for a countable monoid $G\subset X^X$, $X$ admits a non-discrete Hausdorff $G$-topology if and only if the Zariski $G$-topology $\zeta_G$ is non-discrete; moreover, in this case, $X$ admits $2^{\mathfrak c}$ hereditarily normal $G$-topologies.Comment: 10 page

Algebraically determined topologies on permutation groups

In this paper we answer several questions of Dikran Dikranjan about algebraically determined topologies on the group of (finitely supported) permutations of a set X.Comment: 10 page