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.

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.

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


  1. Rudin, W., Principles of Mathematical Analysis, Third Edition, 1976, McGraw-Hill, Inc, New York.

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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