On Cellular-Lindelöf Spaces.

In this paper, we make several observations on cellular-Lindelöf spaces. We prove that in perfect spaces, the property of being cellular-Lindelöf is equivalent to the countable chain condition. Using this result, we prove that every cellular-Lindelöf first-countable perfect space has cardinality at most c and obtain a regular example of a weakly Lindelöf non-cellular-Lindelöf space. We also prove that if X is a cellular-Lindelöf space then every discrete family of non-empty open subsets of X is countable. Finally, we prove that if X is a cellular-Lindelöf space with a symmetric g-function such that ∩{g2(n,x):n∈ω}={x} for each x∈X then |X|≤2c. [ABSTRACT FROM AUTHOR]