  We discuss the Becchi-Rouet-Stora-Tyutin (BRST) cohomology and Hodge
decomposition theorem for the two dimensional free U(1) gauge theory. In
addition to the usual BRST charge, we derive a local, conserved and nilpotent
co(dual)-BRST charge under which the gauge-fixing term remains invariant. We
express the Hodge decomposition theorem in terms of these charges and the
Laplacian operator. We take a single photon state in the quantum Hilbert space
and demonstrate the notion of gauge invariance, no-(anti)ghost theorem,
transversality of photon and establish the topological nature of this theory by
exploiting the concepts of BRST cohomology and Hodge decomposition theorem. In
fact, the topological nature of this theory is encoded in the vanishing of the
Laplacian operator when equations of motion are exploited. On the two
dimensional compact manifold, we derive two sets of topological invariants with
respect to the conserved and nilpotent BRST- and co-BRST charges and express
the Lagrangian density of the theory as the sum of terms that are BRST- and
co-BRST invariants. Mathematically, this theory captures together some of the
key features of both Witten- and Schwarz type of topological field theories.
