If we could seek shelter inside a domed city, it might be interesting to watch the outside world freeze to zero kelvin. But then if the Sun were to go out, would it be a neutron star or a black hole, or would its mass just vanish?
When the sun does go out (in about 5 billion more years) it will have depleted enough hydrogen for gravity to cause the star to collapse to a white dwarf. It will still continue to radiate heat at a decreased rate for tens of billions of years after that, though. There's also the part about the red giant stage but that's for another post
:^)