User:Caliburn/s/fa/spectral permanence


 * Boundary of Proper Non-Empty Set in Connected Topological Space is Non-Empty
 * Complement of Closed Disk in Complex Plane is Connected
 * Complement of Bounded Set in Complex Plane has at most One Unbounded Component
 * Resolvent Set of Element in Banach Algebra has same Unique Unbounded Component as Resolvent Set of Element in Closed Subalgebra
 * Boundary of Spectrum of Element in Closed Subalgebra of Banach Algebra is contained in Boundary of Spectrum in whole Banach Algebra
 * Spectral Permanence Theorem