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An \xmm observation of the plerionic supernova remnant 3C58 has allowed us tostudy the X-ray nebula with unprecedented detail. A spatially resolved spectralanalysis with a resolution of 8\arcsec has yielded a precise determination ofthe relation between the spectral index and the distance from the center. We donot see any evidence for bright thermal emission from the central core. Incontrast with previous ASCA and {\em Einstein} results, we derive an upperlimit to the black-body 0.5-10 keV luminosity and emitting area of $1.8\times10^{32}$ \ergsec and $1.3\times 10^{10}$ cm$^2$, respectively, ruling outemission from the hot surface of the putative neutron star and also excludingthe "outer-gap" model for hot polar caps. We have performed for the first timea spectral analysis of the outer regions of the X-ray nebula, where most of theemission is still non-thermal, but where the addition of a soft (kT=0.2-0.3keV) optically thin plasma component is required to fit the spectrum at $E<1$keV. This component provides 6% of the whole remnant observed flux in the0.5-10.0 keV band. We show that a Sedov interpretation is incompatible with theSN1181-3C58 association, unless there is a strong deviation from electron-ionenergy equipartition, and that an origin of this thermal emission in terms ofthe expansion of the nebula into the ejecta core nicely fits all the radio andX-ray observations.Comment: 10 pages, 7 figures, accepted for publication in A&

The X-ray nebula of the filled center supernova remnant 3C 58 and its interaction with the environment