  We study Sunyaev-Zeldovich Effect (SZE) cluster counts in different
cosmologies. It is found that even without the full knowledge of the redshift
distribution of SZE clusters, one can still readily distinguish a flat universe
with a cosmological constant from an open universe. We divide clusters into a
low redshift group (with redshift z<0.5) and a high redshift group (with z>1),
and compute the relation r=N(z<0.5)/N(z>1), where N is the number of
flux-limited (S_lim) SZE clusters. With about the same total number of SZE
clusters N(z>0), the r-value for a flat universe with a consmological constant
and that for an open universe occupy different regions in the S_lim -- r plot
for the most likely cosmological parameters 0.25<\Omega_0<0.35 and
0.2<\Gamma<0.3, where \Gamma is the shape parameter of the initial power
spectrum of density fluctuations. Thus, with a deep SZE cluster survey, the
ratio r can reveal, independent of the normalization of the power spectrum,
whether we are living in a low-density flat universe or an open universe.
  Within the flat universe scenario, the SZE cluster-normalized \sigma_8 has
also been investigated in this work, where sigma_8 is the r.m.s. density
fluctuations within the top-hat scale of 8 h^{-1} Mpc. A functional relation
sigma_8 \sim \Omega_0^{-0.13} is found. Combining it with the X-ray
cluster-normalized \sigma_8\sim\Omega_0^{-0.52+0.13\Omega_0}, one can put tight
constraints on both \Omega_0 and \sigma_8 simulataneously.
