  In this paper, Floer homology for Lagrangian submanifolds in an open
symplectic manifold given as the complement of a smooth divisor is discussed.
The main new feature of this construction is that we do not make any assumption
on positivity or negativity of the divisor. To achieve this goal, we use a
compactification of the moduli space of pseudo-holomorphic discs into the
divisor complement satisfying Lagrangian boundary condition that is stronger
than the stable map compactification and is inspired by the compactifications
that are used in relative Gromov--Witten theory. This is the first of a series
of three papers, this compactification is introduced and some of its
fundamental properties as a topological space, essential for the definition of
Lagrangian Floer homology, are established.
