As you add or remove neutrons from the nucleus, isotopes are formed

That's true: isotopes are chemical elements with same atomic number (i.e. same number of protons) but different mass number (i.e. different number of protons+neutrons, so if the number of protons is the same, the number of neutrons should be different). 
For instance, hydrogen has three different isotopes: 
- protium, with only one proton in the nucleus
- deuterium, with one proton and one neutron
- tritium, with one proton and two neutrons