The spaces in which any Lindelöf subspaces are CCC

Abstract

In this paper, we first prove that there exists a weakly Lindelöf non-CCC space X which is the union of a countable family of closed discrete subsets of X , which gives a positive answer to Question 3.14 of [11]. By using the same example, we show that a weakly Lindelöf Tychonoff space in which every Lindelöf subspace is CCC may not be CCC, which answers Question 3.13 of [11] negatively. Finally, we prove that |X | ≤ 2^{Δ2 ( X  ) hLc ( X  )} for any star Lindelöf space X , where Δ₂ (X ) is the minimal cardinal κ such that X has a rank 2 G_κ−diagonal and hLc(X ) is the supremum of the cellularities of Lindelöf subspaces of X .