Abstract
The results in this paper answer three questions asked by Noble in [10] and give a partial answer to a question asked by Alster in [1]. We prove that every Alster space is totally Lindelöf and this gives a new characterization of regular Alster spaces. We construct a non-regular totally Lindelöf space that is not Alster and we prove that there exists a Lindelöf P-space that is not Frolík.
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