I have seen the discussion here: $\psi$-irreducibility of Markov Chains.
For the example in the last paragraph where $X=\{a,b\}$ and the Markov chain can only jump from $a$ to $a$ and $b$ to $b$ with probability one, can we let $\mathcal{B}(X)=\{\varnothing, \{a,b\}\}$ with $\varphi(\varnothing)=0$ and $\varphi(\{a,b\})=1$? Does it mean the Markov chain becomes $\varphi$-irreducible?