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Distinguishing Lindelöfness and inverse Lindelöfness. (English) Zbl 0838.54004
A space \(X\) is inversely Lindelöf provided for every open cover \({\mathcal U}\) of \(X\) there exists a countable subset \({\mathcal U}_0 \subset {\mathcal U}\) and a function \(f : {\mathcal U}_0 \to {\mathcal T} (X)\) such that for each \(U \in {\mathcal U}_0\), \(f(U)\) equals either \(U\) or \(X \smallsetminus U\) and \(\{f(U) : U \in {\mathcal U}_0\}\) covers \(X\) (in other words, to choose a countable “subcover”, one can take not only the elements of the given cover, but their complements as well). Other inverse covering properties, such as inverse compactness and inverse countable compactness are defined in a similar way. Clearly, the inverse modification of a covering property is a priori weaker than the original property, but they can turn out to be equivalent. The reviewer has shown that every inversely countably compact space is countably compact [Topology Appl. 62, No. 2, 181-191 (1995; Zbl 0837.54013)], and that every inversely pseudocompact space is pseudocompact. It is an open problem whether there exists a Hausdorff, inversely compact space which is not compact.
The paper under review contains examples of inversely Lindelöf spaces that are not Lindelöf. First of all, the author notes that all spaces of cardinality less than \({\mathfrak c}\) are inversely Lindelöf, hence, assuming the negation of CH, the discrete space of cardinality \(\omega_1\) is inversely Lindelöf, non-Lindelöf. Next, he shows that any Ostaszewski space is inversely Lindelöf, hence inverse Lindelöfness together with (inverse) countable compactness do not imply inverse compactness. Assuming CH, the author constructs a locally countable topology on \(\omega_1\) which is inversely Lindelöf, non-Lindelöf. Since there are both CH- and \(\neg \text{CH}\)-examples, it can be considered a ZFC fact that inverse Lindelöfness does not imply Lindelöfness.
MSC:
54A35 Consistency and independence results in general topology
03E35 Consistency and independence results
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
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