Why define the Cantor set with an intersection? Define $E_n$ as
$ E_1 = \left[0,\frac{1}{3}\right] \cup \left[\frac{2}{3},1\right]$
$ E_2 = \left[0,\frac{1}{9}\right] \cup \left[\frac{2}{9},\frac{3}{9}\right] \cup \left[\frac{6}{9},\frac{7}{9}\right] \cup \left[\frac{8}{9},1\right]$
and so on.
I usually see the Cantor set defined as $C = \bigcap_{n=1}^\infty E_n $. Why use this limit with an intersection, instead of the seemingly more natural $C = \lim_{n\to\infty} E_n $ ?
As far as I can tell, when the limit isn't involved, the intersection is unnecessary:
$E_N = \bigcap_{n=1}^N E_n $
 A: Limits are associated with topology; whereas the first time I saw the Cantor set constructed, I was in my first semester. Topology had yet to come out behind the curtain.
For a freshman, limits are for sequences of numbers, or functions of real and complex numbers. Limits are preserved for objects which are not sets in their essence. On the other hands, when you have a family of sets you obviously can discuss their unions and intersections.
So from a pedagogical point of view, this is indeed a reasonable approach to avoid the discussion about continuous operations on sets in the topological space $\mathcal P(\Bbb R)$ (the power set of the real numbers).
Much much later I have learned that a function from ordinals into sets is called continuous if at limit stages we have $\bigcup_{\alpha<\delta} f(\alpha)=f(\bigcup_{\alpha<\delta}\alpha)=f(\lim_{\alpha\to\delta}\alpha)=f(\delta)$. 
In this aspect, $E_n$ makes somewhat of a continuous sequence of length $\omega$, whose limit point is the Cantor set. The above definition fails here because we wish to discuss intersection and the continuity defined above discussed unions, alas this is not an important matter as we can always talk about $D_n=\mathbb R\setminus E_n$ instead.
To sum up my ramble above, yes it is possible to discuss limits instead of intersections, but the limit is the intersection (at least in the Cantor set case), but from a teaching point of view it is often the case where the Cantor set is introduced the students have a proper grasp of topology, sufficient to discuss limits of sets. In this case the use of intersections which is much clearer to understand when discussing sets is better.
