This paper considers the question of when a space with the property that each discrete subspace is countable is hereditarily Lindelbölf. The question is answered affirmatively for the class of R O P spaces and for the class of hereditarily meta-Lindelöf spaces. A characterization of hereditarily Lindelöf spaces in terms of countable subspaces is given.
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Reilly et al. (1980) studied this question.
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