Adding a back flow preventer or a check valve to a domestic water
supply will create a closed system.

Usually the issue of thermal expansion in a close system is handled by
installing an expansion tank on the cold water side of the water
heater.

Could the thermal expansion in the system be handled by installing a
pressure relief valve; with a relief pressure lower than the relief
pressure of the T/PO valve of the water heater?

Of course this relief valve would dump a small amount of water every
day...... approx the amount of expansion of the cold water heated to
water heater temp.

Is the expansion tank solution "better" than the relief valve
solution? or is a relief valve a better solution?

I'm just curious........... since I can see if the relief valve is
working but the behavior of the expansion tank is internal & thus
hidden.....so I have to go on faith as to whether the expansion tank
is doing it's job.

cheers
Bob