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The combinatorics of open covers. II. (English) Zbl 0870.03021

[For Part I see ibid. 69, No. 1, 31-62 (1996; Zbl 0848.54018).]
Let \(X\) be a space and let \({\mathcal A}\) and \({\mathcal B}\) be families of open covers of \(X\). The authors study the following three topological selection principles for obtaining covers in \({\mathcal B}\) from covers in \({\mathcal A}\):
\(S_1 ({\mathcal A},{\mathcal B})\): For each sequence \(({\mathcal U}_n:n=1,2,3,\dots)\) of elements of \({\mathcal A}\) there is a sequence \((U_n:n =1,2,3,\dots)\) such that for each \(n\), \(U_n\) is an element of \({\mathcal U}_n\), and \((U_n:n=1,2,3,\dots)\) is an element of \(\mathcal B\);
\(S_{\text{fin}}({\mathcal A},{\mathcal B})\): For each sequence \(({\mathcal U}_n:n=1,2,3,\dots)\) of elements of \({\mathcal A}\) there is a sequence \((U_n:n=1,2,3,\dots)\) such that for each \(n\), \(U_n\) is a finite subset of \({\mathcal U}_n\), and \(\bigcup_{n<\infty} U_n\) is an element of \({\mathcal B}\);
\(U_{\text{fin}}({\mathcal A},{\mathcal B})\): For each sequence \(({\mathcal U}_n:n=1,2,3,\dots)\) of elements of \({\mathcal A}\) there is a sequence \((U_n:n=1,2,3,\dots)\) such that for each \(n\), \(U_n\) is a finite subset of \({\mathcal U}_n\), and either there is an \(n\) with \(U_n\) a cover of \(X\), or else \(\{\bigcup U_n:n=1,2,3,\dots\}\) is an element of \({\mathcal B}\).
These three selection principles were inspired by classical literature: the first principle is inspired by a special case of it which was introduced by F. Rothberger [Fundam. Math. 30, 50-55 (1938; Zbl 0018.24701)], the second and the third are both inspired by special cases which were introduced by W. Hurewicz [Math. Z. 24, 401-421 (1925; JFM 51.0454.02)].
The authors consider these selection principles when each of \({\mathcal A}\) and \({\mathcal B}\) is allowed to be one of three special classes of open covers, namely the class of all open covers, the class of \(\omega\)-covers, and the class of \(\gamma\)-covers. Their main results include: At least ten and at most eleven of these classes are nonempty and distinct (for one pair of these classes it is an open problem whether they are distinct); this fact is witnessed by subspaces of the real line (in some cases an additional axiom such as the Continuum Hypothesis is required to show this); a conjecture made in the 1925 paper by Hurewicz [loc. cit.] is false.

MSC:

03E05 Other combinatorial set theory
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
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