[en] The extreme effects of radiative cooling on the structure of an inflow of material into the Galaxy are explored. Significantly, it is found that cooling cannot prevent the formation of a very hot (T approximately 2 x 10⁶ K) corona in the flow close to the Galaxy. It is shown that a supernova rate in the Galaxy of 1 per 50 years is sufficient to generate and maintain throughout the interstellar medium a network of interconnected tunnels containing very low-density gas and occupying at least 20 percent of the overall volume of the disk. These tunnels would have n less than or equal to 10^-2 cm^-3 and T approximately 10⁶ K. The hot, low-density cavities of individual old supernova remnants are found to be enclosed within thick neutral shells, unless such shells are hydrodynamically unstable during the very late phase of remnant evolution. If such a shell is encountered from the outside by the strong shock front generated by another supernova, then the shell quickly becomes Rayleigh--Taylor unstable and fragments as the new shock breaks into the old cavity. Such intersections leading to connections between low-density cavities increase the volume in the cavities and eventually build long tunnels containing material which is kept hot by fast shock waves propagating through them, primarily initiated by more supernova blasts occurring within the tunnels. The tunnel gas has a very long cooling time and loses appreciable heat by thermal conduction only along whatever magnetic field lines connect hot and cold regions. Tunnels are therefore probably destroyed principally by interstellar mass motions. A detailed computer simulation of interacting supernova remnants has been constructed to study this model. Tunnel gas can produce the soft x-ray background flux seen at low galactic latitudes, and the boundary regions between hot tunnel gas and cooler interstellar gas may make a contribution to interstellar lines of O VI