  We study the topology of two-dimensional open systems in terms of the Green's
function. The Ishikawa-Matsuyama formula for the integer topological invariant
is applied in open systems, which indicates the number difference of gapless
edge bands arising from the poles and zeros of the Green's function. Meanwhile,
we define another topological invariant via the single-particle density matrix,
which works for general gapped systems and is equivalent to the former for the
case of weak coupling to an environment. We also discuss two applications. For
time-reversal-invariant insulators, the ${Z}_{2}$ index can be expressed by the
invariant of each spin subsystem. As a second application, we consider the
proximity effect when an ordinary insulator is coupled to a topological
insulator.
