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Register a new userOn meager function spaces, network character and meager convergence in topological spaces
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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 $betaomega$.
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The notion of Haar null set was introduced by J. P. R. Christensen in 1973 and reintroduced in 1992 in the context of dynamical systems by Hunt, Sauer and Yorke. During the last twenty years this notion has been useful in studying exceptional sets in diverse areas. These include analysis, dynamical systems, group theory, and descriptive set theory. Inspired by these various results, we introduce the topological analogue of the notion of Haar null set. We call it Haar meager set. We prove some basic properties of this notion, state some open problems and suggest a possible line of investigation which may lead to the unification of these two notions in certain context.
We show that an ideal $mathcal{I}$ on the positive integers is meager if and only if there exists a bounded nonconvergent real sequence $x$ such that the set of subsequences [resp. permutations] of $x$ which preserve the set of $mathcal{I}$-limit points is comeager and, in addition, every accumulation point of $x$ is also an $mathcal{I}$-limit point (that is, a limit of a subsequence $(x_{n_k})$ such that ${n_1,n_2,ldots,} otin mathcal{I}$). The analogous characterization holds also for $mathcal{I}$-cluster points.
In this paper, we continue the study of function spaces equipped with topologies of (strong) uniform convergence on bornologies initiated by Beer and Levi cite{beer-levi:09}. In particular, we investigate some topological properties these function spaces defined by topological games. In addition, we also give further characterizations of metrizability and completeness properties of these function spaces.
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.
Our main problem is to find finite topological spaces to within homeomorphism, given (also to within homeomorphism) the quotient-spaces obtained by identifying one point of the space with each one of the other points. In a previous version of this paper, our aim was to reconstruct a topological space from its quotient-spaces; but a reconstruction is not always possible either in the sense that several non-homeomorphic topological spaces yield the same quotient-spaces, or in the sense that no topological space yields an arbitrarily given family of quotient-spaces. In this version of the paper we present an algorithm that detects, and deals with, all these situations.
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