Decide if the following sets are manifolds and say why. If there are exceptional points at which the sets are not manifolds, give them:
(a) Phase space of Hamiltonian mechanics, the space of the canonical coordinates and momenta pi and qi;
(b) The interior of a circle of unit radius in two-dimensional Euclidean space;
(c) The set of permutations of n objects;
(d) The subset of Euclidean space of two dimensions (coordinates x and y) which is a solution to xy (x2 + y2 − 1) = 0.