  Evolution of the cluster population has been recognized as a powerful
cosmological tool. While the present--day abundance of X-ray clusters is
degenerate in $\sigma_8$, $n$ and $\Omega_0$, Oukbir and Blanchard (1992, 1997)
have pointed out that the number density evolution of X-ray clusters with
redshift can be used to determine $\Omega_0$. Here, we clarify the origin of
this statement by identifying those parameters to which the evolution of
cluster number density is most sensitive. We find that the evolution is
controlled by only two parameters: the amplitude of fluctuations, $\sigma_M$,
on the scale associated with the mass under consideration, $R = 9.5h^{1/3}
\Omega_0^{-1/3}$ $ M_{15}^{1/3} h^{-1}$Mpc, and the cosmological background
density, $\Omega_0$. In contrast, evolution is remarkably insensitive to the
slope of the power spectrum. We verify that the number density evolution of
clusters is a powerful probe of the mean density of the universe, under the
condition that $\sigma_M$ is chosen to reproduce current-day abundances.
Comparison of the cluster abundance at $z \sim 0.5-0.6$, from the EMSS, to the
present-day abundance, from the ROSAT BCS sample, unambiguously reveals the
existence of significant negative evolution. This number evolution, in
conjunction with the absence of any negative evolution in the
luminosity-temperature relation, provides robust evidence in favor of a
critical density universe ($\Omega_0=1$), in agreement with the analysis by
Sadat et al. (1998).
