Cardinal invariants of dually CCC spaces

We say that a space X is dually CCC (respectively, weakly Lindelof, separable) if for any neighbourhood assignment phi on X, there is a CCC (respectively, weakly Lindelof, separable) subspace Y subset of X such that phi(Y) = {phi(y) : y is an element of Y} covers X. In this paper, we mainly show that (1) A dually CCC first countable Hausdorff space has cardinality at most 2(c) and a dually weakly Lindelof first countable normal space has cardinality at most 2(c). (2) Let Y = Pi{Y-i: i <= n}, where Y-i is a scattered monotonically normal space for any i = 0, 1, ..., n. If a subspace X subset of Y is dually CCC then e(X) <= omega and a normal subspace X subset of Y is DCCC if and only if e(X) <= omega. (3) Assume 2(<c) = c. A normal dually CCC space X with chi(X) <= c has extent at most c. (4) A dually separable Hausdorff space X with a G(delta)*-diagonal has extent at most c and a dually separable regular space X with a G(delta)-diagonal has cardinality at most c. (5) A dually CCC Hausdorff space with a G(delta)-diagonal has cellularity at most c. (C) 2021 Elsevier B.V. All rights reserved.