Notation for the set of all $x_i$ Maybe a silly question, but what is the most compact way of clearly expressing the set of all $x_i$ if it is obvious that $i$ runs from 1 to $n$?  $\{x_i\}$?  $\{x_i\}_i$?  $\{x_i\}_{i=1}^n$?  $\{x_1, x_2, \dotsc, x_n\}$?
 A: The third: $\{x_i\}_{i = 1}^n\;$ would be the most compact, but could open the door for ambiguity. 
For example, $\{a_n\}$ and/or $\{a_n\}_{i = 1}^n$ is sometimes used (unfortunately, for the same reason: ambiguity) to denote a sequence. 
So I'd prefer $$\{x_i\mid i\in \mathbb N,\;1 \leq i \leq n\},$$ as this more clearly denotes the set as defined in your last alternative: $$\{x_1, x_2, x_3, \cdots, x_n\}$$
and doesn't rely on "it being clear that $i$ runs from $1$ to $n$." (Assuming also $n \in \mathbb N$ ...)
A: It depends on what you mean by "the most compact" as there is a trade-off between notation with a short number of strokes to write on a chalkboard and expressive power to avoid confusion. For what I have seen in papers, if the index is the same throughout a proof then it is customary to call $I$ the index and write $$\{x_{i}\}_{i \in I}$$
when there is not a constant index and the index is finite with upper bound $n$ we would write $$\{x_{i}\}_{i=1}^{n}$$
and if there is not a constant index and the index is infinite we write
$$\{x_{i}\}_{i=1}^{\infty}$$
