I am studying my reading material for the markov chains section of my degree. In particular, the section I'm confused about is on the topic of recurrence and transience. There is a theorem, which states,
Intercommunicating states have the same recurrence status. Suppose $i \leftrightarrow j$ and state $i$ is recurrent, then state $j$ is also recurrent.
That's fine and intuitive and I understand that. However, there is a note beneath that which confuses me. It says,
This theorem implies that, if $C$ is an irreducible subset of states, then either the states in $C$ are all recurrent or all transient.
Now I try to reconcile this with the definition of an irreducible set of states. That is,
A set $C$ of states is called irreducible if $i \leftrightarrow j$ $ \forall i, j \in C$.
That means I can pick any state in $C$ and find a way to get back to that state. In other words, there are no transient states. Which to my mind, confirms that "if $C$ is an irreducible subset of states, then the states in $C$ are all recurrent". Therefore, I do not understand how an irreducible set of states can possibly consist of only transient states. Can somebody explain this please? Maybe there is a misunderstanding in my chain of reasoning?
Edit: I think I have figured out why this is by drawing this diagram:
States 1 and 2 are transient, since given enough steps, we will transition to absorbing state 3. But 1 and 2 belong to the irreducible set $C$.
