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Applications of \(k\)-covers. (English) Zbl 1111.54023

An open cover \({\mathcal U}\) of a space \(X\) is called a \(k\)-cover if every compact subset of \(X\) is contained in a member of \({\mathcal U}\) and \(X\) is not a member of \({\mathcal U}\). \({\mathcal K}\) denotes the collection of \(k\)-covers of \(X\).
Let \({\mathcal A}\) and \({\mathcal B}\) be collections of subsets of an infinite set \(X\). Then \(S_{\text{fin}}({\mathcal A},{\mathcal B})\) denotes the selection principle:
For each sequence \((A_n: n\in\mathbb{N})\) of elements of \({\mathcal A}\) there is a sequence \((B_n: n\in\mathbb{N})\) of finite sets such that for each \(n\,B_n\subset A_n\) and \(\bigcup_{n\in\mathbb{N}}B_n\) is an element of \({\mathcal B}\).
\(G_{\text{fin}}({\mathcal A},{\mathcal B})\) is a game corresponding to \(S_{\text{fin}}({\mathcal A},{\mathcal B})\).
The authors investigate the role of \(k\)-covers in selection principles theory and, as one result they prove:
Theorem 7. For a space \(X\) the following are equivalent:
(1) \(X\) satisfies \(S_{\text{fin}}({\mathcal K},{\mathcal K})\); (2) \(X\) satisfies \({\mathcal K}\to [{\mathcal K}]^2_2\). Here \({\mathcal K}\to[{\mathcal K}]^2_k\) denotes the following statement: For each \(A\in{\mathcal K}\) and for each function \(f: [A]^2\to\{1,\dots, k\}\) there are a \(B\in\{B\}\), \(B\subset A\), a \(j\in\{1,\dots,k\}\) and a partition \(B= \bigcup_{n\in\mathbb{N}}B_n\) of \(B\) into finite sets such that for each \(\{a, b\}\in[B]^2\) with \(a\) and \(b\) are not in the same \(B_n\), \(f(\{a, b\})= j\) holds.
An open cover \({\mathcal U}\) of a space \(X\) is called a \(\gamma\)-cover if it is infinite, and each point of \(X\) belongs to all but finitely many elements of \({\mathcal U}\). \(\Gamma\) denotes the collection of \(\gamma\)-covers of \(X\).
The authors prove the following:
Theorem 10. For a space \(X\) the following are equivalent:
(1) \(X\) satisfies \(S_{\text{fin}}({\mathcal K},\Gamma)\); (2) \(X\) satisfies \(S_1({\mathcal K},\Gamma)\); (3) ONE does not have a winning strategy in the game \(G_1({\mathcal K},\Gamma)\) in \(X\).
Relating to Hurewicz property and Menger property the authors introduce selection principles \(U_{\text{fin}}(\Gamma,{\mathcal K})\) and \(U_{\text{fin}}({\mathcal K},\Gamma)\) and show generalized assertions of the \(S_{\text{fin}}\) type, and also consider star selection principles and \(k\)-covers.

MSC:

54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
03E02 Partition relations
91A44 Games involving topology, set theory, or logic
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