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Register a new userOn some functional generalizations of the regularity of topological spaces
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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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The structure of topological spaces is analysed here through the lenses of fibrous preorders. Each topological space has an associated fibrous preorder and those fibrous preorders which return a topological space are called spacial. A special class of spacial fibrous preorders consisting of an interconnected family of preorders indexed by a unitary magma is called cartesian and studied here. Topological spaces that are obtained from those fibrous preorders, with a unitary magma emph{I}, are called emph{I}-cartesian and characterized. The characterization reveals a hidden structure of such spaces. Several other characterizations are obtained and special attention is drawn to the case of a monoid equipped with a topology. A wide range of examples is provided, as well as general procedures to obtain topologies from other data types such as groups and their actions. Metric spaces and normed spaces are considered as well.
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.
For a non-isolated point $x$ of a topological space $X$ the network character $nw_chi(x)$ is the smallest cardinality of a family of infinite subsets of $X$ such that each neighborhood $O(x)$ of $x$ contains a set from the family. We prove that (1) each infinite compact Hausdorff space $X$ contains a non-isolated point $x$ with $nw_chi(x)=aleph_0$; (2) for each point $xin X$ with countable character there is an injective sequence in $X$ that $F$-converges to $x$ for some meager filter $F$ on $omega$; (3) if a functionally Hausdorff space $X$ contains an $F$-convergent injective sequence for some meager filter $F$, then for every $T_1$-space $Y$ that contains two non-empty open sets with disjoint closures, the function space $C_p(X,Y)$ is meager. Also we investigate properties of filters $F$ admitting an injective $F$-convergent sequence in $betaomega$.
We prove that for every $T_0$ space $X$, there is a well-filtered space $W(X)$ and a continuous mapping $eta_X: Xlra W(X)$ such that for any well-filtered space $Y$ and any continuous mapping $f: Xlra Y$ there is a unique continuous mapping $hat{f}: W(X)lra Y$ such that $f=hat{f}circ eta_X$. Such a space $W(X)$ will be called the well-filterification of $X$. This result gives a positive answer to one of the major open problems on well-filtered spaces. Another result on well-filtered spaces we will prove is that the product of two well-filtered spaces is well-filtered.
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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