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Register a new userSome classes of minimally almost periodic topological groups
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Classes SSGP(n)(n < omega) of topological groups are defined, and the class-theoretic inclusions SSGP(n) subseteq SSGP(n+1) subseteq m.a.p. are established and shown proper. These classes are investigated with respect to the properties normally studied by topologists (products, quotients, passage to dense subgroups, and the like). In passing the authors establish the presence of the SSGP(1) or SSGP(2) property in many of the early examples in the literature of abelian m.a.p. groups.
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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.
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.
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 $G$.
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.
We introduce and study some generalizations of regular spaces, which were motivated by studying continuity properties of functions between (regular) topological spaces. In particular, we prove that a first-countable Hausdorff topological space is regular if and only if it does not contain a topological copy of the Gutik hedgehog.
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