I predict the evolution of the X-ray luminosity function of clusters of galaxies. Predominantly, I treat the assumption that galaxies form first, then cluster purely due to gravitation. I show that the richness distribution of Abell clusters favors this scenario, rather than the protocluster hypothesis. The luminosity function is produced by combining a generalized (for all Ω) Press-Schechter evolutionary mass function for clusters (derived herein) with a power law X-ray luminosity-mass relation; a power law relation is supported by observations of low-redshift clusters.

I find very steep evolution in the luminosity function, and thus in the source counts, for large Ω, and moderate evolution for small Ω. For a variety of models for the gas supply rate to the intracluster medium, the evolution of the luminosity function does not vary greatly. Thus it appears that the Ω, dependence will dominate and that number counts of X-ray clusters will yield cosmological information. The power of a test of Ω with an evolving luminosity function is considerably enhanced relative to a test which involves solely global cosmological effects on a non-evolving population. This occurs because of the well-known result that, at late times, clustering tends to proceed slowly for universes of small Ω and rapidly for large Ω.