Assumption: Let $P$ be an arbitrary Markov chain, and let $i, j$ be two distinct states in the state space.
Theorem:
Strict definition of recurrence: State $i$ is recurrent if and only if
$$ f_{ii} = \mathbb{P}_i(T_i^+ < \infty) = 1. $$Asymmetry: The condition
$$ f_{ij} = 1 $$starting from $i$, state $j$ is reached with probability $1$, is not sufficient to imply
$$ f_{ii} = 1. $$A transient chain may still satisfy $f_{ij} = 1$, for example when the path must pass through a particular state before escaping to infinity.