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Summary  

A subset Y of a space X is weakly almost Lindelf in X if for every open cover U of X, there exists a countable subfamily  V of  U such that Y ⊆ comp (∩V ). We investigate the relationship between relatively weakly almost  Lindelf subsets and relatively almost  Lindelf subsets, and also study various properties of relatively weakly almost  Lindelf subsets.

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Abstract

We identify continuous real-valued functions on a Tychonoff space X with their (closed) graphs thus allowing for C(X) to naturally inherit the lower Vietoris topology from the ambient hyperspace. We then calculate a bitopological version of tightness using the weak Lindelöf numbers of finite powers of X. We also characterize bitopological versions of countable fan and strong fan tightness of the point-open topology with respect to the lower Vietoris topology on C(X) in terms of suitable covering properties of the powers X n formulated using the language of S 1 and S fin selection principles.

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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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Society , to appear. Pol, R. , A function space C(X) which is weakly Lindelöf but not weakly compactly generated, Studia Mathematicae , 64 (1979), 279–285. MR 0544732 80j :46042

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