Cardinal invariants of dually CCC spaces

Abstract

We say that a space X is dually CCC (respectively, weakly Lindelöf, separable) if for any neighbourhood assignment ϕ on X, there is a CCC (respectively, weakly Lindelöf, separable) subspace Y⊂X such that ϕ(Y)={ϕ(y):y∈Y} covers X. In this paper, we mainly show that(1) A dually CCC first countable Hausdorff space has cardinality at most 2c and a dually weakly Lindelöf first countable normal space has cardinality at most 2c.(2) Let Y=∏{Yi:i≤n}, where Yi is a scattered monotonically normal space for any i=0,1,...,n. If a subspace X⊂Y is dually CCC then e(X)≤ω and a normal subspace X⊂Y is DCCC if and only if e(X)≤ω.(3) Assume 2<c=c. A normal dually CCC space X with χ(X)≤c has extent at most c.(4) A dually separable Hausdorff space X with a Gδ⁎-diagonal has extent at most c and a dually separable regular space X with a Gδ-diagonal has cardinality at most c.(5) A dually CCC Hausdorff space with a Gδ-diagonal has cellularity at most c.