  Let V be a Euclidean Jordan algebra of rank n. The eigenvalue map from V to
R^n takes any element x in V to the vector of eigenvalues of x written in the
decreasing order. A spectral set in V is the inverse image of a permutation set
in R^n under the eigenvalue map. If the permutation set is also a convex cone,
the spectral set is said to be a spectral cone. This paper deals with
connectedness and arcwise connectedness properties of spectral sets. By relying
on the result that in a simple Euclidean Jordan algebra, every eigenvalue orbit
is arcwise connected, we show that if a permutation invariant set is connected
(arcwise connected), then the corresponding spectral set is connected
(respectively, arcwise connected). A related result is that in a simple
Euclidean Jordan algebra, every pointed spectral cone is irreducible.
