Definition of the Cantor Set

Question

The Cantor set is usually defined in the following way:

Let $A_1 = [0, 1]$ and \begin{equation} A_n = A_{n-1} \setminus \bigcup_{k=1}^\infty \left( \frac{1 + 3k}{3^n}, \frac{2 + 3k}{3^n} \right) \textrm{for}\ n > 1 \textrm{.} \end{equation}

Then, the Cantor Set, $\mathcal{C}$, is \begin{equation} \mathcal{C} = \bigcap_{n \in \mathbb{N}} A_n \textrm{.} \end{equation}

My question is: can't we just define the Cantor Set as \begin{equation} \mathcal{C} = \lim_{n \to \infty} A_n \textrm{.} \end{equation}

Why not? Is it because we haven't already defined what does the limit of such a sequence mean?

Answer 1

Is it because we haven't already defined what does the limit of such a sequence mean?

Yup, that's exactly it!

Intuitively it's clear that - insofar as the limit of a sequence of sets exists - the Cantor set is indeed the limit of the $A_n$s. But until we give a precise definition of the limit of a sequence of sets, that idea can't be used to define the Cantor set.

(Also, let's say we use the following definition: $\lim_{n\rightarrow\infty}S_n=T$ iff $$T=\{x: \exists n\forall m>n(x\in S_m)\},$$ which seems pretty good to me. Then we still can't define the Cantor set as the limit of the $A_n$s - we first need to show that that limit exists in the first place. This amounts to observing that $A_0\supseteq A_1\supseteq ...$ - but then we've more-or-less used rather roundabout language to define the Cantor set as the intersection of the $A_i$s! So in this particular case at least, it seems to just amount to packaging a simple definition in more technical, if snappier, language.)

Answer 2

I remember asking the same exact question when I first took analysis and my professor sighed and said "I was afraid someone was going to ask that."

It doesn't make sense for the simple reason that limits are defined for sequences of real numbers, not sets. Of course, the concept of limits generalizes is many ways, but in a first or second analysis course, limits are only defined for:

• sequences of real numbers (and thus series)
• functions
• (possibly) metric spaces.

That said, the joke's on my professor because I finally learned in grad school that one can define the notion of limits in any category. This is probably not how you should be thinking about it, but I believe that, in the category of sets, inverse limits are given by intersections.

Answer 3

We haven't defined a limit of a sequence of subsets of $[0,\,1]$, because we haven't defined a metric on such sets. By the way, another common definition is $A_{n+1}=\frac13A_n\cup\left(\frac23+\frac13A_n\right)$, where $a+bS:=\{a+bx|x\in S\}$.