  The aims of this article is to generalize some useful Besicovitch-Morse type
covering lemmas in complete Riemannian manifolds and try to find more spaces
such that the so-called BCP and WBCP are equivalent while these two properties
are weaker and still useful. We also get interest in the best constants of
Besicovitch-type covering properties in Euclidean spaces and sorted out the
best results of related problems before giving a new proof of Besicovitch
covering theorem in the one-dimensional case.
