On the Set Version of Selectively Star-CCC Spaces

A space X is said to be set selectively star-ccc if for each nonempty subset B of X, for each collection U of open sets in X such that B¯⊂∪U, and for each sequence An:n∈ℕ of maximal cellular open families in X, there is a sequence An:n∈ℕ such that, for each n∈ℕ, An∈An and B⊂St∪n∈ℕAn,U. In this paper, we introduce set selectively star-ccc spaces and investigate the relationship between set selectively star-ccc and other related spaces. We also study the topological properties of set selectively star-ccc spaces. Some open problems are posed.