Leaves of a regular foliation

Given a regular foliation atlas on $M$, call each connected component of a slice

${\phi }^{-1}(\phi (U)\cap ({\mathbb{R}}^k\times \{c\}))$

an atlas plaque. Declare two points equivalent when they can be joined by a finite chain of atlas plaques in which consecutive plaques intersect. A leaf is an equivalence class for this plaque-chain relation. Thus a leaf is the underlying subset obtained by continuing one local plaque through overlapping foliation charts, not an arbitrary connected union of distinct leaves. This item defines only that underlying subset; it does not yet assign the subset an intrinsic smooth manifold structure.