Remarks on monotone (weak) Lindelöfness

Using Erclos-Rado's theorem, we show that (1) every monotonically weakly Lindelof space satisfies the property that every family of cardinality c(+) consisting of nonempty open subsets has an uncountable linked subfamily; (2) every monotonically Lindelof space has strong caliber (c(+), w1), in particular a monotonically Lindelof space is hereditarily c-Lindelof 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) Lindelof spaces are also discussed. For example, we show that the Pixley-Roy space PR(X) of a space X is monotonically Lindelof if and only if X is countable and every finite power of X is monotonically Lindelof.