On Σ*-spaces and strong Σ*-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 Gδ-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 Gδ-set of X, then X is a strong Σ-space. As a corollary, we have that a T2-space is a σ-space iff it is a Σ*-space with a Gδ-diagonal. Some conditions that imply Σ*-property are discussed.