  We examine how the Sunyaev-Zel'dovich (SZ) effect of clusters can probe
$\Omega_{o}$. Using self-consistent models of X-ray clusters in the context of
hierarchical models of structure formation, we show that both the mean Compton
{\it y} parameter and the number of clusters observed via the SZ effect
strongly depend on $\Omega_{o}$. These quantities are higher in open
cosmological models due to the earlier epoch of structure formation.
Specifically, we compare two models which are able to reproduce the present
abundance of X-ray clusters, one for $\Omega_{o}=0.2$ and the other $\Omega =
1$. For $\Omega_{o}=0.2$, ${\ymean}$ exceeds the current FIRAS limit of
$1.5\times10^{-5}$. Also, the SZ redshift source distribution considerably
differs in the two cosmological models. Thus, these results show that both an
improvement in our knowledge of CMB spectral distortions and the realization of
millimeter surveys to look for SZ sources over a large area of the sky should
produce interesting constraints on $\Omega_{o}$ and on the evolution of the
baryonic fractionof virialized structures.
