The concept of a ∆-set of reals was originally defined by G.M. Reed. An equivalent version was defined by Eric van Douwe and later on was generalized to an arbitrary topology space, (∆-space) . Historically, this notion arose in the study of the normal Moore space conjecture, where Q-sets were used to construct important counterexamples to the conjecture. We prove that Moore's L-space (a hereditarily Lindelöf but not separable space in ZFC) is not a Q-set space and if Aronszajn tree naturally associated with Moore’s L-space is special Moore's L-space will not be a ∆-space.
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Let
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By a space we mean a metrizable separable zero-dimensional space.
A space
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We deal with the class of Hausdorff spaces having a
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Combinatorial covering properties as Rothberger’s, Hurewicz’s and Menger’s are procedures for generating a cover of a given topological space from a sequence of covers of this space.
We present the most celebrated such properties together with the most important examples in a classical countable case. We also explore how these notions and examples extend to the uncountable context, where the initial sequence of covers has length
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Let
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All spaces are assumed to be separable and metrizable. A space X is countable dense homogeneous (CDH) if all countable dense subsets of X can be mapped onto each other by homeomorphisms of X. The fundamental theorem of countable dense homogeneity states that every "sufficiently homogeneous" Polish space is CDH. This result motivated a long-standing search for examples of non-Polish CDH spaces. We contribute to this line of research by exhibiting a non-Polish CDH topological group. This is joint work with Claudio Agostini and Lyubomyr Zdomskyy.
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Frechet spaces are selectively separable. Finite products of countable Frechet spaces need not be Frechet but it is independent as to whether they are selectively separable. We will review some recent results on this topic and, at the moment, think we have a new one.
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Given a metric space and a positive number
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In this talk we will be looking at conditions on a plane continuum
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We study ``metrizaion theorem in forcing extensions.'' That is, for a non-metrizable space
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Following earlier authors in this subject, the topology induced by a modular metric is herein called a modular topology. We show that such a topology is metrizable. More precisely, we show that the uniform topology induced by the uniformity on the modular set of a modular pseudometric is metrizable. In addition, we observe that such a topology is coarser than the underlying topology of the uniformity induced by the corresponding pseudometric. Other related immediate observations are also presented.
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The notion of a partial metric space was introduced by Matthews in 1994 who showed, roughly speaking, how metric--like tools can be extended to non--Hausdorff topologies. He also indicated some applications of this class of spaces in the study of denotational semantics of a programming language. In this talk we are going to present some necessary and sufficient conditions under which the topology generated by a partial metric is equivalent to the topology generated by a suitably defined metric. Next, we are going to focus on two basic topological properties of partial metric spaces, namely completeness and compactness. In particular, it appears that in these spaces compactness is equivalent to sequential compactness. Finally, we will focus on a very general fixed point theorem for mappings acting in partial metric spaces. In that theorem we impose some conditions on behavior of considered mappings on orbits and a condition relating orbits of points of small size.
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This is a joint work with Evgenii Reznichenko.
A topological group is said to be
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This talk's purpose is to present some results on whether a
topological space
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In this talk, we will examine variants of sequential compactness and countable compactness that are associated with Ramsey-like properties. These notions have arisen naturally in various topological and combinatorial contexts. We will present several results obtained by treating these compactness properties as central objects of study. The talk will conclude with some connections to classical problems and open questions.
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All spaces are assumed to be Tychonoff spaces. Let
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We use
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Function spaces have been studied in the theory of infinite-dimensional topology, and their Borel hierarchies play important roles in recognizing topologies on them. In this talk, we shall investigate the Borel hierarchies and the complete metrizability of function spaces consisting of metrics on metrizable spaces, and as an application, we will decide their topological types.
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In his survey article in the Handbook of Set-theoretic Topology on cardinal characteristics of the continuum and small cardinals arising in topology, van Douwen introduced three such invariants of a separable metrizable space, M, namely cof(K(M)), kc(M) and k(M). Each invariant asks for the minimum size of a family of compact subsets of M with certain properties.The third invariant, k(M), requires that the compact subsets witness the k-space property of M. In this talk we aim to understand not just the size, but the "shape" of compact families witnessing the k-space property (k-structures), and the "shape" of families of convergent sequences witnessing sequentiality (sequential structures), of a separable metrizable space. Our primary tool will be an extension, due to Vojtas, of the Tukey order on directed sets to general relations. A natural question arising from this work will have as its answer, `the omega_1 st fixed point of the aleph function'.
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If
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