How to characterize recurrent and transient states of Markov chain

(0) The definition of $T_i$ on Wikipedia is awful on at least three counts. One should define $T_i=\inf A_i$ with $A_i=\{n\ge1:X_n=i\}\cup\{+\infty\}$ and one should say that $i$ is transient if and only if $P(T_i<+\infty|X_0=i)<1$ and recurrent otherwise.

(1) In some cases there exists no closed subset at all and the existence of closed sets is not related to recurrence or transience.

First example: consider a homogenous random walk on $\mathbb{Z}$. Thus $p_{n,n+1}=a$ and $p_{n,n-1}=1-a$ for every $n$, for a given $a$ in $(0,1)$. Then there exists no closed set except $\mathbb{Z}$ and the chain is recurrent if $a=\frac12$ and transient otherwise.

Second example: consider a homogenous birth-and-death chain such that $0$ is absorbing. Thus $p_{0,0}=1$ and $p_{n,n+1}=a$ and $p_{n,n-1}=1-a$ for every $n\ge1$, for a given $a$ in $(0,1)$. Then the set $S=\{0\}$ is closed and the chain is recurrent if $a\le\frac12$ and transient otherwise.

(2) You could (should?) read the beautiful small book Random Walks and Electric Networks by Peter G. Doyle and J. Laurie Snell, which explains this and a lot of related stuff in a very accessible way.

1. Tim's characterization of states in terms of closed sets is correct for finite state space Markov chains. Partition the state space into communicating classes. Every recurrent class is closed, but no transient class is closed (because the chain must eventually get "stuck" in some recurrent class). The part in parentheses is false for infinite state space chains, as Didier's answer shows.

2. Another well-known characterization is that a state $i$ is transient if and only if $$\sum_{n=1}^\infty P(X_n=i | X_0=i)<\infty.$$ This criterion is used, for example, to prove Polya's result that the symmetric random walk on $\mathbb{Z}^d$ is recurrent if $d=1,2$, but transient when $d\geq 3$.

3. Similarly, the probability
   $$P(X_n=i\mbox{ for infinitely many } n | X_0=i)$$ is equal to zero or one, depending on whether the state $i$ is transient or recurrent.