Identifying the communicating classes and stating whether they are period, aperiodic, recurrent or transient

I don't understand what the answers say

The transition matrix is

$$\begin{pmatrix} 0.5 & 0.5 & 0 \\ 0 & 0.5 & 0.5\\ 0 & 0 & 1 \end{pmatrix}$$

$C_1 = \{3\}, T = \{1,2\}$, state 3 is aperiodic because $p_{33} > 0$ and recurrent. States 1 and 2 are aperiodic and transient.

I understand the bits about the individual states, but what does $C_1 = \{3\}, T = \{1,2\}$ mean? The lecturer hasn't explained this or written this in the notes. Is this saying the closed state is 3 and the other communicating classes are 1 and 2?

EDIT: C is the set of irreducible closed classes and T is the set of transient classes

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Are you asking for anything beyond explication of the set theory notation? –  whuber Dec 15 '12 at 21:32

$T = \{1,2\}$ almost certainly means that the set of transient states has as elements the states $1$ and $2$.

$C_1 = \{3\}$ probably means that one of irreducible closed sets has as its only element elements the state $3$. If there were more than one irreducible closed set (not this example) then you might see $C_2$ or others.

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As Henry pointed out, $T$ is the set of transient states only. $C$ contains only one state and it is a closed set. The subscript 1 comes here, because there is only one such closed set.
$p_{33}>0$ means $p_{33}^{1}>0$ (which is the 1st step transition probability). In fact as $p_{33}=1$, so the last row always remains as it is. Thus, $p_{33}^{n}>0$ always for all $n$. So the period, $d(3)=gcd\{n:p_{33}^{n}>0\}=1$