on the triangulated category of perfect complexes on $X$.[^6] So, one may (roughly) say that any “reasonable” weight structure (on a triangulated category satisfying some Brown representability condition) can be used to construct certain $t$-structures. This demonstrates the importance of constructing weight structures.

However, the $t$-structures constructed using Theorem \[tadjti\] appear to be somewhat “exotic” (yet cf. Theorem 1 of [@zvon]) and possibly the $t$-structures constructed via the aforementioned “$R$-saturated” versions of the theorem are “more useful”. Still we also prove (using adjacent and [*$\Phi$-orthogonal*]{} weight structures in a crucial way) several properties of [compactly generated]{} $t$-structures.[^7]

To formulate the following theorem we need some definitions.

For a class of objects $S$ of a triangulated category ${\underline{C}}$ we will write $S{{}^{\perp}}$ (resp. ${{}^{\perp}}S$) for the class of those $M\in {\operatorname{Obj}}{\underline{C}}$ such that the morphism group ${\underline{C}}(N,M)$ (resp. ${\underline{C}}(M,N)$) is zero for all $N\in S$.

We will say that a $t$-structure $t=({\underline{C}}^{t\le 0},{\underline{C}}^{t\ge 0})$ on ${\underline{C}}$ is [*generated*]{} by a class ${\mathcal{P}}\subset {\operatorname{Obj}}{\underline{C}}$ whenever ${\underline{C}}^{t\ge 0}=\cap_{i\ge 1}({\mathcal{P}}{{}^{\perp}}[i])$.

\[tgroth\] Let ${\underline{C}}$ be a triangulated category that is closed with respect to coproducts; let $t$ be a $t$-structure on it generated by a set of compact objects (we will say that $t$ is 