Exponential separability is preserved by some products

[摘要] We show that exponential separability is an inverse invariant of closed maps with countably compact exponentially separable fibers. This implies that it is preserved by products with a scattered compact factor and in the products of sequential countably compact spaces. We also provide an example of a \sigma-compact crowded space in which all countable subspaces are scattered. If X is a Lindelöf space and every Y\subset X with |Y|\leq 2^{\omega_1} is scattered, then X is functionally countable; if every Y\subset X with |Y|\leq 2^{\mathfrak{c}} is scattered, then X is exponentially separable. A Lindelöf \Sigma-space X must be exponentially separable provided that every Y\subset X with |Y|\leq {\mathfrak{c}} is scattered. Under the Luzin axiom ({\mathfrak{c}} ) we characterize weak exponential separability of C_p(X,[0,1]) for any metrizable space X. Our results solve several published open questions.

[学科分类] 物理化学和理论化学

[关键词] Lindelöf space;scattered space;\sigma-product;function space;P-space;exponentially separable space;product;functionally countable space;weakly exponentially separable space