The finer selection properties (Q2087779)

For a function \(f\) from \(\omega\) to \(\omega\), a topological space \(X\) satisfies \(\bigcup_{f}\{\Gamma,\Gamma)\) if for each sequence \(\{\mathcal{U}_{n}:n\in\omega,\, \mathcal{U}_{n}\text{ has no finite subcover}\}\) of elements of \(\Gamma\), select for each \(n\) a finite subset \(\mathcal{F}_{n}\subseteq\mathcal{U}_{n}\) such that \(|\mathcal{F}_{n}|\leq f(n)\) for all \(n\) and \(\{\bigcup\mathcal{F}_{n}:n\in\omega\}\) is an element of \(\Gamma\), where \(\Gamma\) denotes the family of all open \(\gamma\)-covers of \(X\). In this paper, the authors prove the following results. \textbf{Theorem 3.1.} Assume the Continuum Hypothesis. There is a set of real numbers that satisfies \(\bigcup_{\mathrm{fin}}(\Gamma,\Gamma)\) and \(S_{1}(\Gamma,O)\) but not \(\bigcup_{\mathrm{id}}(\Gamma,\Gamma)\), where \(\mathrm{id}\) is the identity function from \(\omega\) to \(\omega\). \textbf{Theorem 4.1} Assume \(\mathfrak{b}=\mathfrak{c}\). There is a set of real numbers that satisfies \(\bigcup_{\mathrm{id}}(\Gamma,\Gamma)\) but not \(\bigcup_{k}(\Gamma,\Gamma)\) for all natural numbers \(k\geq1\). \textbf{Theorem 5.2.} Assume the Continuum Hypothesis. For each natural number \(k\geq2\), there is a set of real numbers that satisfies \(\bigcup_{k+1}(\Gamma,\Gamma)\) but not \(\bigcup_{k}(\Gamma,\Gamma)\).