Random forcing and (S) and (L).
The author gives some conditions for two propositions (S) and (L) to be true or not. (S): Every hereditarily separable space is hereditarily Lindelöf and (L): Every hereditarily Lindelöf space is hereditarily separable. Under \({\mathfrak M}{\mathfrak A}_{\aleph_1}\), after forcing with a separable measure algebra, (S) and (L) are true in the class of cometrizable spaces. A measure is finitely additive, the unit interval \([0,1]\)-valued and strictly positive. For a nonseparable measure algebra, there are counterexamples to (S) and (L). If there is a Sierpiński set, then there is a compact space which contains a counterexample to (L). The Wage lemma \(W(\theta)\) holds after focing with any measure algebra. Under the existence of a Sierpiński set, there is a compact space which contains a counterexample to (L). There are two questions, both under \({\mathfrak M}{\mathfrak A}_{\aleph_1}\). After forcing with any measure algebra, does \(X^2\) contain uncountable discrete subsets whenever \(X\) is nonmetrizable and compact? Another one is: After forcing with any measure algebra, does (S) hold for first countable compact spaces?
\({\mathfrak M}{\mathfrak A}_{\aleph_1}\)
scientific article; zbMATH DE number 1955990
scientific article