  The theory for the formation of the first population of stars (Pop III)
predicts a IMF composed predominantly of high-mass stars, in contrast to the
present-day IMF, which tends to yield stars with masses less than 1 M_Solar.
The leading theory for the transition in the characteristic stellar mass
predicts that the cause is the extra cooling provided by increasing metallicity
and in particular the cooling provided at high densities by dust. The aim of
this work is to test whether dust cooling can lead to fragmentation and be
responsible for this transition. To investigate this, we make use of
high-resolution hydrodynamic simulations. We follow the thermodynamic evolution
of the gas by solving the full thermal energy equation, and also track the
evolution of the dust temperature and the chemical evolution of the gas. We
model clouds with different metallicities, and determine the properties of the
cloud at the point at which it undergoes gravitational fragmentation. We follow
the further collapse to scales of an AU when we replace very dense,
gravitationally bound, and collapsing regions by a simple and nongaseous
object, a sink particle. Our results suggest that for metallicities as small as
10^{-5}Z_Solar, dust cooling produces low-mass fragments and hence can
potentially enable the formation of low mass stars. We conclude that dust
cooling affects the fragmentation of low-metallicity gas clouds and plays an
important role in shaping the stellar IMF even at these very low metallicities.
We find that the characteristic fragment mass increases with decreasing
metallicity, but find no evidence for a sudden transition in the behaviour of
the IMF within the range of metallicites examined in our present study.
