On -spaces and strong -spaces of countable pseudocharacter

Lin raised the following open problem in 1995: If X is a strong Σ * -space and every point of X is a G δ -set of X, is X a strong Σ-space? In this paper, this problem is answered. The main conclusion is that if X is a strong Σ * -space and every point of X is a G δ -set of X, then X is a strong Σ-space. As a corollary, we have that a T 2 -space is a σ-space iff it is a Σ * -space with a G δ -diagonal. Some conditions that imply Σ * -property are discussed.