We observed several nearby face-on spiral galaxies with the ROSAT PSPC to study their 0.1-2.0 keV diffuse emission. After the exclusion of resolved discrete sources, there is unresolved X-ray emission in all the galaxies observed. Since this emission is a combination of diffuse emission and a contribution from unresolved point sources, it represents an upper limit to the truly diffuse soft X-ray emission. The derived upper limits on the diffuse emission can be interpreted in terms of upper limits to the average intensity of a putative hot halo. They can also be used to derived limits to the total energy radiated by hot gas in the observed galaxies as a function of its temperature for various assumed absorbing geometries. Beyond the equivalent solar radius (the radius at which the Sun would be in the observed galaxies), the temperature of hot gas radiating more than 30% of the total supernova power in the galaxies must be less than $10^{6.1} K$ if it is located within the disk with an assumed absorbing overburden of $3times 10^{20} cm^{-2}$, or less than $10^{5.9} K$ if it lies in an unabsorbed halo.
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