  The single-degenerate model for the progenitors of type Ia supernovae (SNe
Ia) is one of the two most popular models, in which a carbon-oxygen white dwarf
(CO WD) accretes hydrogen-rich material from its companion, increases its mass
to the Chandrasekhar mass limit, and then explodes as a SN Ia. Incorporating
the prescription of Hachisu et al. (1999a) for the accretion efficiency into
Eggleton's stellar evolution code and assuming that the prescription is valid
for \emph{all} metallicities, we carried out a detailed binary evolution study
with different metallicities. We show the initial and final parameter space for
SNe Ia in a ($\log P-M_{\rm 2}$) plane. The positions of some famous recurrent
novae in the ($\log P-M_{\rm 2}$) plane, as well as a supersoft X-ray source
(SSS), RX J0513.9-6951 are well explained by our model, and our model can also
explain the space velocity and mass of Tycho G, which is now suggested to be
the companion star of Tycho's supernova . Our study indicates that the SSS, V
Sge, is a potential progenitor of supernovae like SN 2002ic if the delayed
dynamical-instability model in Han & Podsiadlowski (2006) is appropriate.
