In this paper, it is discussed the AP-property of function spaces. We prove that for any compact network α for a space X which is closed under finite unions, (1) if C α (X) is an AP-space and X is paracompact, then X is a Hurewicz space; (2) if C α (X) is an AP-space which has countable tightness, then C α (X) is discretely generated.
It is discussed in this paper the spaces with σ-point-discrete N_0-weak bases. The main results are: (1) A space X has a σ-compact-finite N_0-weak base if and only if X is a k-space with a σ-point-discrete N_0-weak base; (2) Under (CH), every separable space with a σ-point-discrete N_0-weak base has a countable N_0-weak base.
