Why can a Sun-like star leave behind a white dwarf after shedding its outer layers?

Answer: Its exposed core remains supported by the pressure of tightly packed electrons.

Its exposed core remains supported by the pressure of tightly packed electrons. ✓ — The leftover core can resist further collapse through the quantum pressure of tightly packed electrons, called electron degeneracy pressure. It no longer needs ongoing fusion to provide that support. A white dwarf shines mainly by releasing stored heat and gradually cools.

Its cooled core shines only by reflecting light from nearby stars. — White dwarfs give off their own stored thermal energy rather than relying on reflected starlight. A newly exposed core can be extremely hot even after its main fusion reactions have ended. It dims as that heat escapes over time.

Its expelled gas gathers back into a new star and restarts fusion. — The white dwarf is the old star’s exposed core, not expelled gas that has gathered into a newborn star. Its usual glow comes from residual heat rather than renewed fusion. New stars form by a different process, when interstellar gas clouds contract.