## 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:

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