Transition from countable to uncountable

cantor set

The Cantor set $\mathcal C$ is created by iteratively deleting the open middle third from a set of line segments. Let $\mathcal C_n$ be the set after n iteration. The common definition of Cantor set is the intersection of $\mathcal C_n$ for all n.

  1. Can we define the Cantor set as S=limit $\mathcal C_n$ when n goes to infinity? If not, what is the difference of S and $\mathcal C$? Is S countable?

  2. For each n, $\mathcal C_n$ is composed of countably many closed sets. How can a countable intersection of these sets (for all n) become uncountably many closed sets (points)?

  3. The complement of $\mathcal C_n$ is the union of countably many open intervals. Same question can be asked about the complement of $\mathcal C$. How can the complement of $\mathcal C$ becomes union of uncountably many open intervals?

What are the general conclusions? Is the countable union of countable sets countable? (Consider the set of the midpoints of intervals removed from $\mathcal C_n$.)

Best Answer

Can we define the Cantor set as $S = \lim C_n$ when n goes to infinity?

Yes, but for somewhat trivial reasons: the way the limit of a sequence of nested sets $C_1 \supseteq C_2 \supseteq C_3 \supseteq \cdots$ is defined is as the intersection $$ \lim_{n\to \infty} C_n = \bigcap_{n \in \mathbb N} C_n, $$ so you can define the Cantor set this way, but it's the exact same definition with different notation.

How can the intersection of these sets (for all n) become uncountably many closed sets (points)?

This is a question that's somewhat impossible to answer. How? Just look at the construction of the Cantor set, and you can see it happening. There is something of an implication in this question that it is weird that an intersection of countably many sets which are unions of finitely many closed sets would give a closed set with uncountably many components. Perhaps that is weird, but it's certainly not in contradiction with anything (as this example shows).

How can the complement of $C$ becomes union of uncountably many open intervals?

In fact, the complement of $C$ is also the union of countably many open intervals; this despite the fact that there appear to be "uncountably many gaps in $C$". Perhaps this is causing you confusion: it's similar to the fact that in between any two (of uncountably many) irrationals, there is a rational (out of countably many).

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