A study of selectively star-ccc spaces

Abstract

Bal and Kočinac in [2] introduced and studied the class of selectively star-ccc spaces. A space X is called selectively star-ccc if for every open cover U of X and every sequence (An:n∈ω) of maximal pairwise disjoint open families in X there exists a sequence (An:n∈ω) such that An∈An for every n∈ω and St(⋃n∈ωAn,U)=X. In this paper, we prove that:(1) A selectively star-ccc space is DCCC and a selectively star-ccc perfect space is CCC.(2) There exists a pseudocompact (hence, DCCC) space that is not selectively star-ccc.(3) Every selectively star-ccc subspace of the product of finitely many scattered monotonically normal spaces has countable extent.(4) Every selectively star-ccc subspace of ω1ω has countable extent.(5) Under 2ℵ0=2ℵ1, there exists a selectively star-ccc normal space having a regular closed Gδ-subset which is not selectively star-ccc.(6) Every first countable selectively star-ccc space with a Gδ-diagonal has cardinality at most c.(7) Every selectively star-ccc space with a rank 2-diagonal has cardinality at most c.