Can a subset of the indexing set of a RV, given rise to a non-stationary time series, be considered weakly stationary or vice versa? Given a non-stationary time series $X_t$, is it possible that a subset $[t_1, t_2]$ of the indexing set is considered weakly stationary? And also, given a weakly stationary time-series, can a subset of its indexing set be considered non-stationary?
Thanks :)
EDIT: As pointed out by @Chris Haug, @Richard Hardy I was originally thinking about subsets of the support of the RV that generated the series, rather than about subsets of the indexing set. However, as the answer by @Dilip Sarwate made clear, that type of definition would not be directly related to weakly stationary processes anymore, but would need to refer to a different sort of classification.
Since I haven't seen any duplicate questions, I leave the question up as an answer to the indexing-set question though. Thanks everyone!

As Chris Haug says, the answer to the second question "Can a subset of a weakly stationary time series be nonstationary?" is No.
A time series is modeled as a discrete-time random processes, that is as a collection of indexed random variables $\{X(n)\colon n \in \mathbb T\}$ where the index set $\mathbb T$ is a subset of $\mathbb Z$, the set of all integers. A typical $\mathbb T$ is the set $\{0,1, 2, \ldots, N-1\}$.  The time series is called weakly stationary if it satisfies two criteria (Note: both criteria must be satisfied):

*

*The mean of the time series is a constant, that is, $E[X(k)]$ has the same value $\mu$ for all $k \in \mathbb T$.

*Given any fixed integer $m \in \mathbb Z$, $E[X(k)X(k+m)]$ has the same value $R_X(m)$ for all $k$ such that $k, k+m \in \mathbb T$. Note that the value of $E[X(k)X(k+m)]$ is dependent on $m$ but is not a function of $k$ at all.

The function $R_X(m)$ is called the autocorrelation function of the time series, and has the properties that $|R_X(m)| \leq R_X(0)$. That is, the autocorrelation function has a peak at the origin.
So, how can a weakly stationary time series be nonstationary, say between $k_1$ and $k_2$? If the mean varies from the fixed value $\mu$ that it enjoys for $k<k_1$ and $k>k_2$, Criterion 1 is not satisfied, and so the assumption of weak stationarity falls to the ground. If for some $k \in \{k_1,k_1+1,k_1+2 \ldots, k_2\}$, it so happens that $E[X(k)X(k+m)]$ has value that depends on both $k$ and $m$, or is not equal to $R_X(m)$, Criterion 2 is not satisfied, and the claim of weak stationarity is no longer valid.
If a time series is to be called weakly stationary, it must satisfy the two criteria for all time instants such that $k \in \mathbb T$ and $k+m \in \mathbb T$. Of course, the OP can always coin a new name (say Schroederian weak stationarity or weakly stationary in the Schroederian sense) for a time series that is weakly stationary most of the time but fails to be weakly stationary for certain time instants or for certain time intervals, and if the notion is useful and leads to new insights, others will adopt it too.
Turning to the first question, we see that it is just the mirror image of the second question. A time series that is weakly stationary in the Schroederian sense can always be regarded as a nonstationary time series that happens to satisfy the criteria for weak stationarity every now and then.
