## 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.)

### Citations:

Zbl 0848.54018; Zbl 0018.24701; JFM 51.0454.02
Full Text:

### References:

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