The possibility of using quantum sets in quantum mechanics was considered long ago. In this book we consider fractal Cantorian sets and emphasize the role of the empty set in physics and the crucial difference between the zero and the empty set. The quantum particle is represented by a Cantorian zero set while the quantum wave is represented by an empty Cantorian set. The new quantum entanglement theory is presented generalizing Einstein's energy equation to an effective quantum gravity formula predicting accurately the actually measured cosmic energy content of the universe. The energy given by Einstein's famous formula consists of two parts. The first part is the ordinary energy of the quantum particle modelled by the topology of the zero set. The second part is the dark energy of the quantum wave modelled by the topology of the empty set.