Let $M$ be a smooth manifold with foliation $F$. I can understand the quotient $M/F$, i.e. $x\sim y$ if and only if $x,~y$ are in the same leaf.
By the Richardson's paper, Transverse manifold and $G$-manifold, the author often writes the notation $M/\bar F$, quotient the closure of the leaves for Riemannian foliation.
Q I do not understand the definition of $M/\bar F$. My guess is that $x\sim y$ if and only if there is a point $p\in M$, such that $x,~y$ locates on the closure of the leaf $\bar{\mathcal F_p}$ passing through $p$. But is it the equivalent relation, i.e. If $x, y \in \bar{\mathcal F_p}$ and $y,~z\in \bar{\mathcal F_{p'}}$, then we have that $x,~z\in \bar{\mathcal F_q}$ for some point $q\in M$?
Or the quotient is well-defined for some special foliation, e.g. fibration?