Cardinal invariants of cellular-Lindelöf spaces

Abstract

A space X is said to be cellular-Lindelöf if for every cellular family U there is a Lindelöf subspace L of X which meets every element of U. Cellular-Lindelöf spaces generalize both Lindelöf spaces and spaces with the countable chain condition. Solving questions of Xuan and Song, we prove that every cellular-Lindelöf monotonically normal space is Lindelöf and that every cellular-Lindelöf space with a regular G δ -diagonal has cardinality at most 2 c . We also prove that every normal cellular-Lindelöf firstcountable space has cardinality at most continuum under 2 <c = c and that every normal cellular-Lindelöf space with a G δ -diagonal of rank 2 has cardinality at most continuum.2000 Mathematics Subject Classification. Primary: 54A25, 54D20; Secondary: 54E99, 54D55.

A note on rank 2 diagonals

“…We define a neighbourhood assignment φ for X as follows: if x ∈ S let φ(x) = St(x, U n 0 ) and if x ∈ X \ S let φ(x) = X \ S. Since X is dually weakly Lindelöf, there exists a weakly Lindelöf subspace Y such that X = {φ(y) : y ∈ Y }. By the way φ is defined, it follows that Related results for the classes of dually ccc spaces and for that of cellular-Lindelöf spaces were proved in [16] and [6].…”

On cellular-countably compact spaces

“…Bella and Spadaro proved that every normal cellular-Lindel öf space X with a G δ -diagonal of rank 2 has cardinality at most c (see [5,Theorem 13]). We have a related result for cellular-countably compact spaces.…”

“…Bella and Spadaro proved that every cellular-Lindel öf space with a regular G δ -diagonal has cardinality at most 2 c (see [5]). Theorem 4.4.…”

“…Indeed in [3] the authors showed that the cardinality of a cellular-Lindel öf first-countable space does not exceed 2 c and asked whether it is always bounded by the continuum. Bella and Spadaro managed to find such a common generalization by other means (see [5]) but the original question is still open despite several attacks by various authors (see, for example, [2,5,12,14]). Moreover, the introduction of the cellular-Lindel öf property led several authors to study cellular-P-spaces, for various other properties P (see, for example, [1,14]).…”