  This paper is dedicated to the study of smashing weight structures (one may
say that these are weight structures "coherent with arbitrary coproducts"), and
the application of their properties to $t$-structures. In particular, we prove
that hearts of compactly generated $t$-structures are Grothendieck abelian;
this statement strengthens earlier results of several other authors.
  The central theorem of the paper is as follows: any perfect set of objects of
a triangulated category generates a weight structure; we say that weight
structures obtained this way are perfectly generated. An important family of
perfectly generated weight structures are the ones adjacent to compactly
generated $t$-structures; they give injective cogenerators for the hearts of
the latter. We also establish the following not so explicit result: any
smashing weight structure on a well generated triangulated category (this class
of categories contains compactly generated ones) is perfectly generated;
actually, we prove more than that.
  Moreover, we give a classification of compactly generated torsion theories
(these generalize both weight structures and $t$-structures) that extends the
corresponding result of D. Pospisil D. and J. \v{S}\v{t}ovi\v{c}ek to arbitrary
smashing triangulated categories.
