Definition (Absorbing set)

A set $C$ is called absorbing if $$P(i,C)=P_{iC}=1, \ \forall i\in C$$ i.e. the probability of settling into a set $C$ in the long run is 1.

Definition (Indistinguishable)

Let $\{ X_{i} \}_{i=1}^{\infty}$ be a MC. If the state space $\mathbb{X}$ does not contain two disjoint s, i.e. $$A\cap B\not=\emptyset, \ \forall A,B\subset \mathbb{X}$$ where $A,B$ are absorbing sets, then $\{ X_{i} \}_{i=1}^{\infty}$ is indistinguishable.