According to the definition of stochastic process in Wikipedia:
Given a probability space $(\Omega, \mathcal{F}, P)$, a stochastic process (or random process) with state space $X$ is a collection of $X$-valued random variables indexed by a set $T$ ("time"). That is, a stochastic process $F$ is a collection
$\{ F_t : t \in T \}$
where each $F_t$ is an $X$-valued random variable.
Does that mean for all values of the index $t$, the state space $X$ and the $\sigma$-algebra on it must be same? For a stochastic process, Can it be allowed that the state space and the $\sigma$-algebra on it is different for different value of the index?
Is it correct that, as a stochastic process, "a random walk on a graph" seems to have different state spaces at different stages/times?
Thanks and regards!