Perfect sets and Cantor sets, II

This is post #11 of the series on the Euclidean topology of the real line. See the links at the bottom for other posts in the series.

In the previous post Perfect sets and Cantor sets, I, we show that every nonempty perfect set is uncountable. We now show that any perfect contains a Cantor set. Hence the cardinality of any perfect set is continuum.

Given a perfect set W, we construct a Cantor set within W. Consider the following cases:

  • Case 1. W contains some bounded closed interval [a,b].
  • Case 2. W does not any bounded closed interval.

If case 1 holds, then we can apply the middle third process on [a,b] and produce a Cantor set. So in the remaining dicussion of this post, we assume case 2 holds. This means that for each closed interval [a,b], there is some x \in [a,b] such that x \notin W.

Background Discussion
Let A \subset \mathbb{R} and p \in A. The point p is a right-sided limit point of A if for each open interval (a,b) containing p, the open interval (p,b) contains a point of A. The point p is a left-sided limit point of A if for each open interval (a,b) containing p, the open interval (a,p) contains a point of A. The point p is a two-sided limit point of A if it is both a right-sided limit point and a left-sided limit point of A. For the proof of the following lemma, see the post labeled #10 listed below.

In Lemma 2 below, we apply the least upper bound property and the greatest lower bound property. See the post labeled #4 listed below.

Key Lemmas for Construction

Lemma 1
Suppose that X \subset \mathbb{R} is an uncountable set. Then X contains a two-sided limit point.

As a corollary to the lemma 1, for the perfect set W in question, all but countably many points of W are two-sided limit points of W.

Lemma 2
Suppose E \subset \mathbb{R} is a nonempty perfect set that satisfies Case 2 indicated above. Suppose that for the closed interval [a,b], we have:

  • (a,b) \cap E \ne \phi,
  • the left endpoint a is a right-sided limit point of E,
  • the right endpoint b is a left-sided limit point of E.

Then we have a<a^*<b^*<b such that:

  • there are no points of E in the open interval (a^*,b^*),
  • the point a^* is a left-sided limit point of E,
  • the point b^* is a right-sided limit point of E.

Proof. Since Case 2 holds, for the closed interval [a,b] in question, there is some x \in (a,b) such that x \notin E. Then we can find an open interval (c,d) such that x \in (c,d) and a<c<d<b and (c,d) \cap E = \phi.

Any point in (c,d) is an upper bound of W_1=[a,c) \cap E. By the least upper bound property, W_1 has a least upper bound a^*. Any point in (c,d) is an lower bound of W_2=(d,b] \cap E. By the greatest lower bound property, W_2 has a greatest lower bound b^*. Then a^* and b^* satisfy the conclusion of the lemma. \blacksquare

Lemma 3
Suppose E \subset \mathbb{R} is a nonempty perfect set. Suppose we have a closed interval [s,t] such that the left endpoint s is a right-sided limit point of E and the right endpoint t is a left-sided limit points of E. Then we have s<s_*<t_*<t such that:

  • the open interval (s_*,t_*) contains points of E,
  • both endpoints s_* and t_* are two-sided limit points of E,
  • t_*-s_*<0.5(t-s).

Proof. Suppose we have a closed interval [s,t] as described in the lemma. Then E_1=[s,t] \cap E is a nonempty perfect set. Thus E_1 is uncountable. So pick p \in (s,t) such that p is a two-sided limit point of E_1.

Choose open interval (c,d) such that s<c<p<d<t and d-c<0.5(t-s). Since p is a two-sided limit point of E_1, choose s^* and t^* such that c<s^*<p, p<t^*<d and both s^* and t^* are two-sided limit points of E_1. It follows that s^* and t^* satisfy the conclusion of the lemma. \blacksquare

Lemma 4
Suppose E \subset \mathbb{R} is a nonempty perfect set that satisfies Case 2 indicated above. Suppose we have a closed interval [a,b] such that the endpoints are two-sided limit points of E. Then we have disjoint closed intervals K_0=[p_0,q_0] and K_1=[p_1,q_1] such that

  • K_0 \subset [a,b] and K_1 \subset [a,b],
  • the lengths of both K_0 and K_1 are less then 0.5(b-a),
  • for each of K_0 and K_1, the endpoints are two-sided limit points of E.

Proof. This is the crux of the construction of a Cantor set from a perfect set and is the result of applying Lemma 2 and Lemma 3.

Applying Lemma 2 on [a,b] and obtain the open interval (a^*,b^*). We remove the open interval (a^*,b^*) from [a,b] and obtain two disjoint closed intervals [a,a^*] and [b^*,b]. Each of these two subintervals contains points of the perfect set W since the endpoints are limit points of W in the correct direction.

Now apply Lemm 3 to shrink [a,a^*] to obtain a smaller subinterval K_0=[p_0,q_0] such that the length of K_0 is less than 0.5(a^*-a) and is thus less than 0.5(b-a). Likewise, apply Lemma 3 on [b^*,b] to obtain K_1=[p_1,q_1] such that the length of K_1 is less than 0.5(b-b^*) and is thus less than 0.5(b-a). Note that both K_0 and K_1 constain points of E, making both K_0 \cap E and K_1 \cap E perfect sets and compact sets. \blacksquare

Construction
Suppose W \subset \mathbb{R} is a nonempty perfect set that satisfies Case 2. Pick two two-sided limit a_0 and b_0 of W. Obtain B_0=K_0 and B_1=K_1 as a result of applying Lemma 4. Let A_1=B_0 \cup B_1.

Then we apply Lemma 4 on the closed interval B_0 and obtain closed intervals B_{00}, B_{01}. Likewise we apply Lemma 4 on the closed interval B_1 and obtain closed intervals B_{10}, B_{11}. Let A_2=B_{00} \cup B_{01} \cup B_{10} \cup B_{11}.

Continue this induction process. Let C=\bigcap \limits_{n=1}^{\infty} A_n. The set C is a Cantor set and has all the properties discussed in the posts labled #6 and #7 lised at the end of this post.

We claim that C \subset W. For any countably infinite sequence g of zeros and ones, let g_n be the first n terms in g. Let y \in C. Then \left\{y\right\}=\bigcap \limits_{n=1}^{\infty} B_{g_n} for some countably infinite sequence g of zeros and ones (see post #6 listed below). Then every open interval (a,b) containing y would contain some closed interval B_{g_n}. Thus y is a limit point of W. Hence y \in W.

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
9. Perfect sets and Cantor sets, I
10. The Lindelof property of the real line

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.