Reeb components of a codimension-one foliation

Assume Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $F$ be a codimension-one regular foliation of a $3$-manifold $M$, and let $X={\bar{D}}^2\times S^1$ be the closed solid torus with its standard Reeb foliation $F_{\mathrm{R}\mathrm{e}\mathrm{e}\mathrm{b}}$ (The Reeb foliation of the solid torus has the boundary as a leaf): the boundary $\mathrm{\partial }X$ is a compact leaf diffeomorphic to $T^2$ and every interior leaf is a plane accumulating on $\mathrm{\partial }X$. A Reeb component of $F$ is a saturated subset $R\subseteq M$, compact as a subset, for which there is a homeomorphism $\mathrm{\Phi }:X\to R$ mapping leaves of $F_{\mathrm{R}\mathrm{e}\mathrm{e}\mathrm{b}}$ onto leaves of $F{\mathrm{\mid }}_R$ and $\mathrm{\partial }X$ onto a leaf of $F$; equivalently, $R$ is compact and saturated, homeomorphic to the solid torus, has a single compact leaf as boundary, and its interior foliation is foliated-homeomorphic to the Reeb foliation. In the smooth models of this page the conjugacy may be taken smooth; in the general closed-leaf construction of lem-the-compact-leaf-produced-by-a-vanishing- cycle-bounds-a-reeb-component the source produces a foliated homeomorphism, so the topological form of the definition is the one used. A foliation with no Reeb component is called Reebless.