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Covering properties of Cp(X) and Ck(X)

Let X be a Tychonoff space. We survey some classic and recent results that
 characterize the topology or cardinality of X when Cp (X) or Ck (X) is
 covered by certain families of sets (sequences, resolutions,
 closure-preserving coverings, compact coverings ordered by a second
 countable space) which swallow or not some classes of sets (compact sets,
 functionally bounded sets, pointwise bounded sets) in C(X).