Let $P_{ij}$ a transition matrix, a class $C$ is closed if given two different states $i$ and $j$ $$i\in C, i\rightarrow j\Rightarrow j\in C$$
If a Markov Chain is irreducible the transition matrix have no closed communicating class, right?
What I do not understand this exercise, it seems to me that the first part contradicts the second. Because if I found a transition matrix with no closed communicating class how I can show that every transition matrix has at least one closed communicating class.
The closed communicating class might be the whole set.
For the second part, you will need an infinite state-space.