Remarks on monotone (weak) Lindelöfness

Using Erdös-Rado's theorem, we show that (1) every monotonically weakly Lindelöf space satisfies the property that every family of cardinality c+ consisting of nonempty open subsets has an uncountable linked subfamily; (2) every monotonically Lindelöf space has strong caliber (c+,ω1), in particular a monotonically Lindelöf space is hereditarily c-Lindelöf and hereditarily c-separable. (1) gives an answer of a question posed in Bonanzinga, Cammaroto and Pansera [3], and (2) gives partial answers of questions posed in Levy and Matveev [15]. Some other properties on monotonically (weakly) Lindelöf spaces are also discussed. For example, we show that the Pixley-Roy space PR(X) of a space X is monotonically Lindelöf if and only if X is countable and every finite power of X is monotonically Lindelöf.