Perfect sets and Cantor sets, I

This is post #9 of the series on the Euclidean topology of the real line. See the links at the bottom for other posts in the series.

Recall that a subset A of the real line is a perfect set if A is closed in Euclidean topology of the real line and that every point of A is a limit point of A. Any closed and bounded interval [a,b] is a perfect set. The Cantor sets (the middle third version and other variations) are perfect sets (see the links #7 and #8 below). It turns out that any nonempty perfect set contains a Cantor set. In this series of posts on Euclidean topology of the real line, by Cantor sets we mean any set that can be constructed by a binary process of splitting closed intervals into two halves at each stage (see links for #6 and #8 below). We demonstrate the algorithm of constructing a Cantor set from any perfect set. This post (part I) shows that any nonempty perfect set is uncountable. Knowing that a perfect is uncountable will simplify the construction process (next post).

Suppose W \subset \mathbb{R} is a nonempty perfect subset. We show that W is uncountable. Since W has at least one point and every point is a limit point, W is infinite. The key to showing W is uncountable is that every nested decreasing sequence of compact subsets of the real line (actually in any topological space) has nonempty intersection. If W happens to be countable, we can define a nested sequence of compact subsets of W with empty intersection. Thus W cannot be countable.

The following lemma is a corollary to Theorem 3 in the post # 4 listed below. The lemma applies to any abstract spaces where compactness can be defined. We state the lemma in terms of the real line since this is our focus.

Lemma
Suppose C_1,C_2,C_3, \cdots are compact subsets of the real line such that

\displaystyle C_1 \supset C_2 \supset C_3 \supset \cdots.

Then \bigcap \limits_{n=1}^{\infty} C_n \ne \phi.

To make the argument that W is uncountable more precise, suppose that W is countable. Then we can enumerate W in a sequence indexed by the positive integers. We have:

\displaystyle W=\left\{w_1,w_2,w_3,\cdots\right\}

Pick a bounded open interval O_1 such that w_1 \in O_1. Next, pick an open interval O_2 such that \overline{O_2} \subset O_1 and w_2 \notin \overline{O_2} and O_2 \cap W \ne \phi.

In the n^{th} stage where n \ge 2, pick an open interval O_n such that \overline{O_n} \subset O_{n-1} and w_n \notin \overline{O_n} and O_n \cap W \ne \phi. Since W is a perfect set, the induction step can continue at every stage.

Now, let C_n=\overline{O_n} \cap W. Note that C_n is a compact set since \overline{O_n} is compact. By the lemma, the intersection of the C_n must be nonempty. By the induction steps, no point of W belongs to all the sets \overline{O_n}, implying the intersection of C_n is empty, a contradiction. Thus W must be uncountable.

Remark
As a corollary, the real line and the unit intervals are uncountable. A more interesting corollary is that any nonempty perfect set has a two-sided limit point. In fact all but countably many points of a nonempty perfect set are two sided limit points. See the post The Lindelof property of the real line for a proof that any uncountable subset of the real line has a two sided limit point. This fact will simplify the construction of a Cantor set from a perfect set.

Links to previous posts on the topology of the real line:
1. The Euclidean topology of the real line (1)
2. The Euclidean topology of the real line (2)
3. The Euclidean topology of the real line (3) – Completeness
4. The Euclidean topology of the real line (4) – Compactness
5. The Cantor bus tour
6. The Cantor set, I
7. The Cantor set, II
8. The Cantor set, III

Reference

  1. Rudin, W., Principles of Mathematical Analysis, Third Edition, 1976, McGraw-Hill, Inc, New York.
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One thought on “Perfect sets and Cantor sets, I

  1. Interesting proof, but I have a question, does exist a direct proof of the fact “perfect sets are uncountable”? I mean, show a bijection between a perfect set X and real numbers.

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