When a chain is irreducible (so each state can be reached from every other state, eventually), we quote that all states have the same character: all aperiodic / periodic with the same period, all transient , all recurrent , all null, all positive, or all ergodic.
Results of this type are called solidarity theorems.
I quote the above from lecture notes. However, I dont understand how can a Chain be irreducible, and have all states transient?
It really doesnt make sense to me. Can someone clarify this for me please ?