  We introduce topological notions of normal crossings symplectic divisor and
variety and establish that they are equivalent, in a suitable sense, to the
desired geometric notions. Our proposed concept of equivalence of associated
topological and geometric notions fits ideally with important constructions in
symplectic topology. This partially answers Gromov's question on the
feasibility of defining singular symplectic (sub)varieties and lays foundation
for rich developments in the future. In subsequent papers, we establish a
smoothability criterion for symplectic normal crossings varieties, in the
process providing the multifold symplectic sum envisioned by Gromov, and
introduce symplectic analogues of logarithmic structures in the context of
normal crossings symplectic divisors.
