A topological space is Suslin ( Lusin ) if it is a continuous (and bijective) image of a Polish space. For a Tychonoff space X let $$C_p(X)$$ C p ( X ) , $$C_k(X)$$ C k ( X ) and $$C_{{\downarrow }{\mathsf {F}}}(X)$$ C ↓ F ( X ) be the space of continuous real-valued functions on X , endowed with the topology of pointwise convergence, the compact-open topology, and the Fell hypograph topology, respectively. For a metrizable space X we prove the equivalence of the following statements: (1) X is $$\sigma $$ σ -compact, (2) $$C_p(X)$$ C p ( X ) is Suslin, (3) $$C_k(X)$$ C k ( X ) is Suslin, (4) $$C_{{\downarrow }{\mathsf {F}}}(X)$$ C ↓ F ( X ) is Suslin, (5) $$C_p(X)$$ C p ( X ) is Lusin, (6) $$C_k(X)$$ C k ( X ) is Lusin, (7) $$C_{{\downarrow }{\mathsf {F}}}(X)$$ C ↓ F ( X ) is Lusin, (8) $$C_p(X)$$ C p ( X ) is $$F_\sigma $$ F σ -Lusin, (9) $$C_k(X)$$ C k ( X ) is $$F_\sigma $$ F σ -Lusin, (10) $$C_{{\downarrow }{\mathsf {F}}}(X)$$ C ↓ F ( X ) is $$C_{\delta \sigma }$$ C δ σ -Lusin. Also we construct an example of a sequential $$\aleph _0$$ ℵ 0 -space X with a unique non-isolated point such that the function spaces $$C_p(X)$$ C p ( X ) , $$C_k(X)$$ C k ( X ) and $$C_{{\downarrow }{\mathsf {F}}}(X)$$ C ↓ F ( X ) are non-Suslin.

中文翻译：

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函数空间的 Lusin 和 Suslin 性质

如果拓扑空间是波兰空间的连续（和双射）图像，则拓扑空间是 Suslin ( Lusin )。对于 Tychonoff 空间 X 让 $$C_p(X)$$C p ( X ) , $$C_k(X)$$ C k ( X ) 和 $$C_{{\downarrow }{\mathsf {F}}} (X)$$ C ↓ F ( X ) 是 X 上的连续实值函数空间，分别具有点收敛拓扑、紧开拓扑和 Fell 超图拓扑。对于可度量空间 X，我们证明以下陈述的等价性：(1) X 是 $$\sigma $$ σ -compact，(2) $$C_p(X)$$ C p ( X ) 是 Suslin，(3 ) $$C_k(X)$$ C k ( X ) 是苏斯林， (4) $$C_{{\downarrow }{\mathsf {F}}}(X)$$ C ↓ F ( X ) 是苏斯林， (5) $$C_p(X)$$ C p ( X ) 是 Lusin， (6) $$C_k(X)$$ C k ( X ) 是 Lusin， (7) $$C_{{\downarrow }{ \mathsf {F}}}(X)$$ C ↓ F ( X ) 是 Lusin， (8) $$C_p(X)$$ C p ( X ) 是 $$F_\sigma $$ F σ -Lusin，(9) $$C_k(X)$$ C k ( X ) 是 $$F_\sigma $$ F σ -Lusin, (10) $$C_{{\downarrow }{\mathsf {F}}}(X )$$ C ↓ F ( X ) 是 $$C_{\delta \sigma }$$ C δ σ -Lusin。我们还构造了一个连续的 $$\aleph _0$$ ℵ 0 -space X 的例子，它有一个唯一的非孤立点，使得函数空间 $$C_p(X)$$ C p ( X ) , $$C_k( X)$$ C k ( X ) 和 $$C_{{\downarrow }{\mathsf {F}}}(X)$$ C ↓ F ( X ) 是非苏斯林。