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Register a new userFree Boolean Topological Groups
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Authors
Olga Sipacheva
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Known and new results on free Boolean topological groups are collected. An account of properties which these groups share with free or free Abelian topological groups and properties specific of free Boolean groups is given. Special emphasis is placed on the application of set-theoretic methods to the study of Boolean topological groups.
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It is proved that any countable topological group in which the filter of neighborhoods of the identity element is not rapid contains a discrete set with precisely one nonisolated point. This gives a negative answer to Protasovs question on the existence in ZFC of a countable nondiscrete group in which all discrete subsets are closed. It is also proved that the existence of a countable nondiscrete extremally disconnected group implies the existence of a rapid ultrafilter and, hence, a countable nondiscrete extremally disconnected group cannot be constructed in ZFC.
A topological group $G$ is called an $M_omega$-group if it admits a countable cover $K$ by closed metrizable subspaces of $G$ such that a subset $U$ of $G$ is open in $G$ if and only if $Ucap K$ is open in $K$ for every $KinK$. It is shown that any two non-metrizable uncountable separable zero-dimenisional $M_omega$-groups are homeomorphic. Together with Zelenyuks classification of countable $k_omega$-groups this implies that the topology of a non-metrizable zero-dimensional $M_omega$-group $G$ is completely determined by its density and the compact scatteredness rank $r(G)$ which, by definition, is equal to the least upper bound of scatteredness indices of scattered compact subspaces of $G$.
A Hausdorff topological group is called minimal if it does not admit a strictly coarser Hausdorff group topology. This paper mostly deals with the topological group $H_+(X)$ of order-preserving homeomorphisms of a compact linearly ordered connected space $X$. We provide a sufficient condition on $X$ under which the topological group $H_+(X)$ is minimal. This condition is satisfied, for example, by: the unit interval, the ordered square, the extended long line and the circle (endowed with its cyclic order). In fact, these groups are even $a$-minimal, meaning, in this setting, that the compact-open topology on $G$ is the smallest Hausdorff group topology on $G$. One of the key ideas is to verify that for such $X$ the Zariski and the Markov topologies on the group $H_+(X)$ coincide with the compact-open topology. The technique in this article is mainly based on a work of Gartside and Glyn.
The famous Banach-Mazur problem, which asks if every infinite-dimensional Banach space has an infinite-dimensional separable quotient Banach space, has remained unsolved for 85 years, though it has been answered in the affirmative for reflexive Banach spaces and even Banach spaces which are duals. The analogous problem for locally convex spaces has been answered in the negative, but has been shown to be true for large classes of locally convex spaces including all non-normable Frechet spaces. In this paper the analogous problem for topological groups is investigated. Indeed there are four natural analogues: Does every non-totally disconnected topological group have a separable quotient group which is (i) non-trivial; (ii) infinite; (iii) metrizable; (iv) infinite metrizable. All four questions are answered here in the negative. However, positive answers are proved for important classes of topological groups including (a) all compact groups; (b) all locally compact abelian groups; (c) all $sigma$-compact locally compact groups; (d) all abelian pro-Lie groups; (e) all $sigma$-compact pro-Lie groups; (f) all pseudocompact groups. Negative answers are proved for precompact groups.
We discuss various modifications of separability, precompactnmess and narrowness in topological groups and test those modifications in the permutation groups $S(X)$ and $S_{<omega}(X)$.
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