Lusin and Suslin properties of function spaces (Q2185010)

For a topological space \(X\) let \(C(X)\) denote the family of all continuous real-valued functions on \(X\). The authors study the descriptive properties of the spaces \(C_p(X)\), \(C_k (X)\) and \(C_{\downarrow F}(X)\), where \(X\) is a Tychonoff space, and \(\bullet\) \(C_p(X)\) denotes the space \(C(X)\) endowed with the topology of pointwise convergence; \(\bullet\) \(C_k(X)\) is the space \(C(X)\) with the the compact-open topology; \(\bullet\) \(C_{\downarrow F}(X)\) is the space \(C(X)\) with the \textit{Fell hypograph topology}, see [\textit{R. A. McCoy} and \textit{I. Ntantu}, Boll. Unione Mat. Ital., VII. Ser., B 6, No. 3, 507--532 (1992; Zbl 0793.54012)]. The main results of the paper are collected in the following Theorem 1.7. Consider the following statements. (1) \(X\) is \(G\)-quotient; (2) \(C_k (X)\) is Polish; (3) \(C_{\downarrow F}(X)\) is \(C_\sigma\)-Lusin; (4) \(X\) is \(F_\sigma\)-quotient; (5) \(C_p(X)\), \(C_k(X)\) are \(F_\sigma\)-Lusin and \(C_{\downarrow F}(X)\) is \(C_{\delta\sigma}\)-Lusin; (6) \(C_p(X)\) is Lusin and \(X\) is an \(\aleph_0\)-space; (7) \(C_k(X)\) is Lusin; (8) \(C_{\downarrow F}(X)\) is Lusin; (9) \(C_p(X)\) is Suslin and \(X\) is an \(\aleph_0\)-space; (10) \(C_k(X)\) is Suslin; (11) \(C_{\downarrow F}(X)\) is Suslin; (12) \(X\) is \(\sigma\)-compact. Then \((3)\Leftarrow (2) \Leftrightarrow (1) \Rightarrow (4) \Rightarrow (5) \Rightarrow (6) \Leftrightarrow (7) \Leftrightarrow (8) \Rightarrow (9) \Leftrightarrow (10) \Leftrightarrow (11) \Rightarrow (12)\). This theorem implies that, if \(X\) is a metrizable space, then the following conditions are equivalent: (1) \(X\) is \(\sigma\)-compact; (2) \(C_p(X)\) is \(F_\sigma\)-Lusin; (3) \(C_k(X)\) is \(F_\sigma\)-Lusin; (4) \(C_{\downarrow F}(X)\) is \(C_{\delta\sigma}\)-Lusin; (5) \(C_p(X)\) is Lusin; (6) \(C_k(X)\) is Lusin; (7) \(C_{\downarrow F}(X)\) is Lusin; (8) \(C_p(X)\) is Suslin; (9) \(C_k(X)\) is Suslin; (10) \(C_{\downarrow F}(X)\) is Suslin.