Remarks on monotone (weak) Lindelöfness (Q529102)

A family of sets is linked provided every two members have non-empty intersection and a space has property \(K(\mathfrak c^+,\omega_1)\) if every cardinality \(\mathfrak c^+\) family of non-empty open subsets has an uncountable linked subfamily. It is shown that every monotonically weakly Lindelöf (mwL) space has property \(K(\mathfrak c^+,\omega_1)\) and that every monotonically Lindelöf (mL) space has calibre \((\mathfrak c^+,\omega_1)\). These results and corollaries are then applied to obtain specific results about a variety of spaces including the one-point Lindelöfication of an uncountable discrete space, GO-spaces (a mL GO-space has weight at most \(\mathfrak c\)), the Alexandroff duplicate (if it is mwL then the original space has cardinality at most \(\mathfrak c\)) and the Pixley-Roy space (Pixley-Roy of \(X\) is mL if and only if \(X\) is countable and every finite power of \(X\) is mL).