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  • Author or Editor: Vladimir V. Tkachuk x
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Summary We prove that, for any Tychonoff  X, the space Cp(X) is K-analytic if and only if it has a compact cover {Kp: p ? ??} such that Kp subset Kq whenever p,q ? ?? and p =q. Applying this result we show that if Cp(X) is K-analytic then Cp(?X) is K-analytic as well. We also establish that a space Cp(X) is K-analytic and Baire if and only if X is countable and discrete.

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Abstract

We prove that, for any cofinally Polish space X, every locally finite family of non-empty open subsets of X is countable. It is also established that Lindelöf domain representable spaces are cofinally Polish and domain representability coincides with subcompactness in the class of σ-compact spaces. It turns out that, for a topological group G whose space has the Lindelöf Σ-property, the space G is domain representable if and only if it is Čech-complete. Our results solve several published open questions.

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Authors: István Juhász, Vladimir V. Tkachuk and Richard G. Wilson

A space X is weakly linearly Lindelöf if for any family U of non-empty open subsets of X of regular uncountable cardinality κ, there exists a point xX such that every neighborhood of x meets κ-many elements of U. We also introduce the concept of almost discretely Lindelöf spaces as the ones in which every discrete subspace can be covered by a Lindelöf subspace. We prove that, in addition to linearly Lindelöf spaces, both weakly Lindelöf spaces and almost discretely Lindelöf spaces are weakly linearly Lindelöf.

The main result of the paper is formulated in the title. It implies that every weakly Lindelöf monotonically normal space is Lindelöf, a result obtained earlier in [3].

We show that, under the hypothesis 2ω < ω ω, if the co-diagonal Δc X = (X × X) \ΔX is discretely Lindelöf, then X is Lindelöf and has a weaker second countable topology; here ΔX = {(x, x): xX} is the diagonal of the space X. Moreover, discrete Lindelöfness of Δc X together with the Lindelöf Σ-property of X imply that X has a countable network.

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