In particular, the spectrum of a commutative C*-algebra is a locally compact Hausdorff space: In the unital case, i.e. where the C*-algebra has a multiplicative unit element 1, all characters "f" must be unital, i.e. "f"(1) is the complex number one. This excludes the zero homomorphism. So "Â" is closed under weak-* convergence and the spectrum is actually "compact". In the non-unital case, the weak-* closure of "Â" is "Â" ∪ {0}, where 0 is the zero homomorphism, and the removal of a single point from a compact Hausdorff space yields a locally compact Hausdorff space.