  We present a study, based on archival XMM-Newton observations, of the
extended X-ray emission associated with the inner disk of M33. After the
exclusion of point sources with L_X > 2 x 10^{35} erg/s (0.3-6 keV), we
investigate the morphology and spectrum of the residual X-ray emission. This
residual emission has a soft X-ray spectrum which can be fitted with a
two-temperature thermal model, with kT = 0.2 keV and 0.6 keV. The soft X-ray
surface brightness distribution shows a strong correlation with FUV emission,
indicative of a close connection between recent star-formation activity and the
production of soft X-rays. Within 3.5 kpc of the nucleus of M33, the soft X-ray
and FUV surface brightness distributions exhibit similar radial profiles. This
implies that the ratio of the soft X-ray luminosity (0.3-2.0 keV) to the star
formation rate (SFR) per unit disk area remains fairly constant within this
inner disk region. We derive a value for this ratio of 1-1.5 x 10^{39}
(erg/s)/(M_sun/yr), consistent with previous studies. In the same region, the
ratio of soft X-ray luminosity to stellar mass (derived from K-band photometry)
is 4 x 10^{28} erg/s/M_sun, a factor of 5-10 higher than is typical of dwarf
elliptical galaxies, suggesting that 10-20% of the unresolved emission seen in
M33 may originate in its old stellar population. The remainder of the soft
X-ray emission is equally split between two spatial components, one which
closely traces the spiral arms of the galaxy and the other more smoothly
distributed across the inner disk of M33. The former must represent a highly
clumped low-filling factor component linked to sites of recent or ongoing star
formation, whereas the distribution of the latter gives few clues as to its
exact origin.
