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.
