Filter One-Point Sums and Lindelöf Cp-Spaces
Abstract
We give a partial answer to a problem of Hernández-Hernández, Ramírez-Chávez and Rojas-Hernández concerning the Lindelöf property of function spaces over one-point extensions of countable free sums. For a free filter \(\mathcal{F}\) on \(\omega\), let \(L_{\mathcal{F}}\) denote the space of real sequences that converge to zero along \(\mathcal{F}\). A coding lemma identifies the descriptive complexity of this auxiliary space by showing that \(L_{\mathcal{F}}\) is \(K\)-analytic if and only if \(\mathcal{F}\) is analytic as a subspace of \(2^\omega\). If \(\mathcal{F}\) is analytic and \((X_n)_{n<\omega}\) is a sequence of non-empty compact spaces, then\[ C_p\left(\bigoplus_{n<\omega} X_n\right)\text{ is Lindelöf} \quad\Longrightarrow\quad C_p(X_{\mathcal{F}})\text{ is Lindelöf}. \]Moreover, if \(C_p(X_{\mathcal{F}})\) is Lindelöf, then \(C_p\left(\bigoplus_{n\in\omega\setminus A} X_n\right)\) is Lindelöf for every \(A\in\mathcal{F}\). Consequently, for constant compact fibres and analytic free filters, \(C_p(X_{\mathcal{F}}(K))\) is Lindelöf and only if \(C_p(K)^\omega\) is Lindelöf. If the free sum has a countable network, no descriptive-set-theoretic restriction on the filter is needed.
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