1. The problem statement, all variables and given/known data As the title says, one needs to show that if A is a closed subspace of a Lindelöf space X, then A is itself Lindelöf. 3. The attempt at a solution Let U be an open covering for the subspace A. (An open covering for a set S is a collection of open sets whose union equals S, btw some define a cover for a set S as a collection such that S is contained in the union of these sets, it seems this causes disambiguity sometimes?) Since all elements of U are open in A, they equal the intersection of some family of open sets with A, call it U'. Now, consider the open cover for X consisting of X\A and U'. By hypothesis, this cover has a countable subcover. Dismiss X\A from it, and then the intersection of the elements left in this collection with A form a countable open cover for A. I hope this works.