  We have used a deep Chandra observation of the central regions of the
twin-jet FRI radio galaxy 3C31 to resolve the thermal X-ray emission in the
central few kiloparsecs of the host galaxy, NGC 383, where the jets are thought
to be decelerating rapidly. This allows us to make high-precision measurements
of the density, temperature and pressure distributions in this region, and to
show that the X-ray emitting gas in the centre of the galaxy has a cooling time
of only 5 times 10^7 years. In a companion paper these measurements are used to
place constraints on models of the jet dynamics.
  A previously unknown one-sided X-ray jet in 3C31, extending up to 8 arcsec
from the nucleus, is detected and resolved. Its structure and steep X-ray
spectrum are similar to those of X-ray jets known in other FRI sources, and we
attribute the radiation to synchrotron emission from a high-energy population
of electrons. In situ particle acceleration is required in the region of the
jet where bulk deceleration is taking place.
  We also present X-ray spectra and luminosities of the galaxies in the Arp 331
chain of which NGC 383 is a member. The spectrum and spatial properties of the
nearby bright X-ray source 1E 0104+3153 are used to argue that the soft X-ray
emission is mostly due to a foreground group of galaxies rather than to the
background broad absorption-line quasar.
