The paper presents a new X-ray-selected and flux-limited sample of galaxy clusters, compiled from the ROSAT All-Sky Survey. The sample, named HIFLUGCS, consists of 63 clusters with galactic latitudes \( |b_{\mathrm{lt}}| \geq 20 \mathrm{deg} \) and X-ray flux \( f_{X}(0.1-2.4 \mathrm{keV}) \geq 2 \times 10^{-11} \mathrm{ergs} \mathrm{s}^{-1} \mathrm{cm}^{-2} \). Gravitational masses are determined using intracluster gas density profiles and gas temperatures, assuming hydrostatic equilibrium. The sample is used to establish the X-ray luminosity–gravitational mass relation and to determine the galaxy cluster mass function. The mass function is constrained to yield cosmological parameters \(\Omega_{\mathrm{m}}=0.12_{-0.04}^{+0.06}\) and \(\sigma_{8}=0.96_{-0.12}^{+0.15}\) (90\% confidence level). Systematic uncertainties are quantified, leading to an upper limit \(\Omega_{\mathrm{m}}<0.31\). The mass function is integrated to show that the mass bound within virialized cluster regions contributes a small fraction to the total matter density, \(\Omega_{\text {Cluster }}=0.012_{-0.004}^{+0.003}\) for clusters with masses greater than \(6.4_{-0.6}^{+0.7} \times 10^{13} h_{50}^{-1} M_{\odot}\).The paper presents a new X-ray-selected and flux-limited sample of galaxy clusters, compiled from the ROSAT All-Sky Survey. The sample, named HIFLUGCS, consists of 63 clusters with galactic latitudes \( |b_{\mathrm{lt}}| \geq 20 \mathrm{deg} \) and X-ray flux \( f_{X}(0.1-2.4 \mathrm{keV}) \geq 2 \times 10^{-11} \mathrm{ergs} \mathrm{s}^{-1} \mathrm{cm}^{-2} \). Gravitational masses are determined using intracluster gas density profiles and gas temperatures, assuming hydrostatic equilibrium. The sample is used to establish the X-ray luminosity–gravitational mass relation and to determine the galaxy cluster mass function. The mass function is constrained to yield cosmological parameters \(\Omega_{\mathrm{m}}=0.12_{-0.04}^{+0.06}\) and \(\sigma_{8}=0.96_{-0.12}^{+0.15}\) (90\% confidence level). Systematic uncertainties are quantified, leading to an upper limit \(\Omega_{\mathrm{m}}<0.31\). The mass function is integrated to show that the mass bound within virialized cluster regions contributes a small fraction to the total matter density, \(\Omega_{\text {Cluster }}=0.012_{-0.004}^{+0.003}\) for clusters with masses greater than \(6.4_{-0.6}^{+0.7} \times 10^{13} h_{50}^{-1} M_{\odot}\).