  We analyze the phemomenon of induced fermion number at finite temperature. At
finite temperature, the induced fermion number $<N>$ is a thermal expectation
value, and we compute the finite temperature fluctuations, $(\Delta
N)^2=<N^2>-<N>^2$. While the zero temperature induced fermion number is
topological and is a sharp observable, the finite temperature induced fermion
number is generically nontopological, and is not a sharp observable. The
fluctuations are due to the mixing of states inherent in any finite temperature
expectation value. We analyze in detail two different cases in 1+1 dimensional
field theory: fermions in a kink background, and fermions in a chiral sigma
model background. At zero temperature the induced fermion numbers for these two
cases are very similar, but at finite temperature they are very different. The
sigma model case is generic and the induced fermion number is nontopological,
but the kink case is special and the fermion number is topological, even at
finite temperature. There is a simple physical interpretation of all these
results in terms of the spectrum of the fermions in the relevant background,
and many of the results generalize to higher dimensional models.
