I'm having a trouble about how to show $T$ is a stopping time. For instance, suppose $X$ is a simple random walk on $I = \mathbb{Z}/k\mathbb{Z}$, and $T$ is the first time $n$ such that $X_0, \dots, X_n$ cover entirely $I$ ($T = \infty$ if never happen).
By following the definition I got We say that $T$ is a stopping time if for all $n \in N$, $\{T ≤ n\}$ only depends on $(X_0, \dots , X_n)$, i.e. there exists a set $A_n \subset I_{n+1}$ such that $\{T ≤ n\} = {(X_0,\dots, X_n) \in A_n}$.
My question is, if $n=k$, we have $X_0, \dots X_n$ cover entirely $I$, but $X_{n+1}=X_0$, does it make $T$ a stopping time?
Another question is, if $T$ is finite, does the probability of $X_{T+2}=X_T$ is $1/k$?
Thank you very much in advance for your help.