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

Just, Winfried; Miller, Arnold W.; Scheepers, Marion; Szeptycki, Paul J.

Topology Appl. 73, No. 3, 241-266 (1996).

[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.

Reviewer: M.Scheepers (Boise)

MSC:

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

Keywords:

diagonalization properties; sequences of open covers; Hurewicz conjecture; \(\omega\)-covers; continuum hypothesis; separable metric spaces; partition properties; Sierpiński set; Lusin set; \(\gamma\)-covers; topological selection principles; JFM 51.0454.02; subspaces of the real line

Citations:

Zbl 0848.54018; Zbl 0018.24701; JFM 51.0454.02

References:

[1]	Bartoszynski, T.; Scheepers, M., A-sets, Real anal. exchange, 19, 2, 521-528, (1993-1994) · Zbl 0822.03028
[2]	Bell, M.; Kunen, K., On the π character of ultrafilters, C. R. math. rep. acad. sci. Canada, 3, 351-365, (1981) · Zbl 0475.54001
[3]	Daniels, P., Pixley-roy spaces over subsets of the reals, Topology appl., 29, 93-106, (1988) · Zbl 0656.54007
[4]	Fremlin, D.H.; Miller, A.W., On some properties of Hurewicz, Menger and rothberger, Fund. math., 129, 17-33, (1988) · Zbl 0665.54026
[5]	Galvin, F.; Miller, A.W., Γ-sets and other singular sets of reals, Topology appl., 17, 145-155, (1984) · Zbl 0551.54001
[6]	Gerlits, J.; Nagy, Zs., Some properties of C(X) I, Topology appl., 14, 151-161, (1982) · Zbl 0503.54020
[7]	Hurewicz, W., Über eine verallgemeinerung des borelschen theorems, Math. Z., 24, 401-421, (1925) · JFM 51.0454.02
[8]	Hurewicz, W., Über folgen stetiger funktionen, Fund. math., 9, 193-204, (1927) · JFM 53.0562.03
[9]	W. Just, On direct sums of γ7 spaces, Manuscript.
[10]	W. Just, γ2 does not imply γ7, Manuscript.
[11]	Lelek, A., Some cover properties of spaces, Fund. math., 64, 209-218, (1969) · Zbl 0175.49603
[12]	Menger, K., Einige überdeckungssätze der punktmengenlehre, Sitzungsberichte. abt. 2a, Mathematik, astronomie, physik, meteorologie und mechanik (Wiener akademie), 133, 421-444, (1924) · JFM 50.0129.01
[13]	Miller, A.W., Special subsets of real line, (), 201-233
[14]	Rothberger, F., Eine verschärfung der eigenschaft C, Fund. math., 30, 50-55, (1938) · JFM 64.0622.01
[15]	Sakai, M., Property C and function spaces, (), 917-919 · Zbl 0691.54007
[16]	Scheepers, M., Combinatorics of open covers I: Ramsey theory, Topology appl., 69, 31-62, (1996) · Zbl 0848.54018
[17]	Sierpiński, S., Sur le produit combinatoire de deux ensembles jouissant de la propriétéC, Fund. math., 24, 48-50, (1935) · JFM 61.0637.05
[18]	W. Stamp, Details supporting “γ2 does not imply γ7“ and “On direct sums of γ7 spaces”, Notes.
[19]	S. Todorčević, Aronszajn orderings, Preprint.
[20]	Vaughan, J.E., Small uncountable cardinals and topology, (), 195-218