  We use the extensive public archive of ROSAT High Resolution Imager (HRI)
observations to carry out a statistical investigation of the X-ray properties
of nearby galaxies. Specifically we focus on the sample of 486 bright (B_T <
12.5) northern galaxies studied by Ho, Filippenko and Sargent (HFS) in the
context of their exploration of the optical spectroscopic properties of nearby
galactic nuclei. Over 20% of HFS galaxies are encompassed in ROSAT HRI fields
of reasonable (> 10ks) exposure. The X-ray sources detected within the optical
extent of each galaxy are categorised as either nuclear or non-nuclear
depending on whether the source is positioned within or outside of a 25
arcsecond radius circle centred on the optical nucleus. A nuclear X-ray source
is detected in over 70% of the galaxies harbouring either a Seyfert or LINER
nucleus compared to a detection rate of only ~40% in less active systems. The
correlation of the H alpha luminosity with nuclear X-ray luminosity previously
observed in QSOs and bright Seyfert 1 galaxies appears to extend down into the
regime of ultra-low luminosity (L(x)~10^38 - 10^40 erg/s) active galactic
nuclei (AGN). The inferred accretion rates for this sample of low-luminosity
AGN are significantly sub-Eddington. In total 142 non-nuclear sources were
detected. In combination with published data for M31 this leads to a luminosity
distribution (normalised to an optical blue luminosity of L(B) = 10^10
L(solar)) for the discrete X-ray source population in spiral galaxies of the
form dN/dL38 = 1.0 +/- 0.2 L38^-1.8, where L38 is the X-ray luminosity in units
of 10^38 erg/s. The implied L(x)/L(B) ratio is ~1.1 x 10^39 erg/s/(10^10
L(solar)). The nature of the substantial number of ``super-luminous''
non-nuclear objects detected in the survey is discussed.
