Question

The following constructs the Cantor set in \(\mathbb{R}\) : begin with the unit interval \([0,1] .\) The set \(F_{1}\) is obtained from this set by removing the middle third, so that $$ F_{1}=\left[0, \frac{1}{3}\right] \cup\left[\frac{2}{3}, 1\right] $$ Now \(F_{2}\) is obtained from \(F_{1}\) by removing the middle third from each of the constituent intervals of \(F_{1} .\) Hence $$ F_{2}=\left[0, \frac{1}{9}\right] \cup\left[\frac{2}{9}, \frac{1}{3}\right] \cup\left[\frac{2}{3}, \frac{7}{9}\right] \cup\left[\frac{8}{9}, 1\right] $$ In general, \(F_{n}\) is the union of \(2^{n}\) intervals, each of which has the form $$ \left[\frac{k}{3^{n}}, \frac{k+1}{3^{n}}\right] $$ where \(k\) lies between 0 and \(3^{n} .\) The Cantor set \(F\) is what remains after this process has been carried out for all \(n\) in \(\mathbb{N}\). Show that (i) \(F\) is closed in \(\mathbb{R}\) (ii) int \(F\) is empty (iii) \(F\) contains no non-empty open set. (iv) the complement of \(F\) can be expressed as a countable union of open intervals. (If you are unfamiliar with the definition of a countable set, then Appendix A gives a brief introduction to this concept.)

Short answer

Question: Prove four properties of the Cantor set: (i) it is closed, (ii) its interior is empty, (iii) it contains no non-empty open set, and (iv) its complement can be expressed as a countable union of open intervals. Answer: (i) The Cantor set is closed since its complement is an open set, which is the union of all removed open intervals during its construction. (ii) The interior of the Cantor set is empty because for any point in the Cantor set, there is always some open interval around it that is removed or not in the Cantor set. (iii) The Cantor set contains no non-empty open set, as the construction of the Cantor set involves successively removing open intervals from the original interval [0,1]. (iv) The complement of the Cantor set can be expressed as a countable union of open intervals, as it is the union of all removed intervals across each step in the construction of the Cantor set.

Step-by-step solution

(i) Proving that the Cantor set is closed

To prove that the Cantor set \(F\) is closed, we need to show that its complement is open. That is, we need to show that for any point \(x\) in the complement of \(F\), there exists an open interval around \(x\) that is completely contained in the complement of \(F\). Let \(x\) be a point in the complement of the Cantor set. This means that \(x\) does not belong to any of the intervals removed during the construction of \(F\). Therefore, all intervals containing \(x\) have been removed, either during the construction of \(F_{n}\) for some \(n\) or in the Cantor set \(F\). Since all the removed intervals are open intervals and the complement of \(F\) is their union, it follows that the complement of \(F\) is an open set. Therefore, \(F\) is closed in \(\mathbb{R}\).

(ii) Proving that the interior of the Cantor set is empty

The interior of the Cantor set consists of all the points of the Cantor set that have an open interval around them completely contained within the Cantor set. Since the construction of the Cantor set involves successively removing open intervals from the original closed interval \([0,1]\), for any point \(x\) in the Cantor set, there will always be an open interval around \(x\) which is removed or not in the Cantor set. Therefore, the Cantor set does not contain any points with an open interval around them that is completely contained within the Cantor set. Hence, the interior of the Cantor set is empty.

(iii) Proving that the Cantor set contains no non-empty open set

If the Cantor set were to contain a non-empty open set, this would imply that there exists an open interval completely contained within the Cantor set. However, this is not possible because the construction of the Cantor set involves successively removing open intervals from the original interval \([0,1]\). Therefore, for any point in the Cantor set, there is always some open interval around it that is removed or not in the Cantor set. Thus, the Cantor set contains no non-empty open set.

(iv) Proving that the complement of the Cantor set can be expressed as a countable union of open intervals

Recall that at each step \(n\) in the construction of the Cantor set, we remove \(2^{n-1}\) open intervals with length \(\frac{1}{3^n}\). Therefore, the complement of the Cantor set is the union of all removed intervals across each step. Since this is a countable union (indexed by \(\mathbb{N}\)) of countable sets (the removed intervals have a bijection with the natural numbers), the complement of the Cantor set can be expressed as a countable union of open intervals.

Key concepts

Abstract Analysis

In the realm of abstract analysis, the Cantor set presents a quintessential example of a counterintuitive construct that challenges our innate perceptions of size and continuity in mathematics. The Cantor set is formed through an iterative process of deletion, where successively smaller middle thirds of intervals are removed from the unit interval on the real number line.To understand this process through the lens of abstract analysis, one must first comprehend the idea of convergence and completeness. The Cantor set is a perfect illustration of a complete metric space, which means that it contains all its limit points, effectively making it a closed set. To further simplify, every sequence of points within the Cantor set that tends to get closer to some value, in fact, has that value as part of the set. Hence, the Cantor set defies the expectation that a set with 'so much taken out' of it could still be so 'full'. As a point of intrigue in abstract analysis, the Cantor set—which is uncountable and has the same cardinality as the set of all reals—exhibits the abstract concept of infinite sets within the rigorous structure of mathematical analysis.

Set Theory

In the framework of set theory, the Cantor set serves as a fascinating object of study, as it encapsulates several fundamental principles such as cardinality, topology, and measure theory. When detailing the Cantor set, we start off with a complete interval and apply an infinite sequence of operations to arrive at the final set, F.

Cardinality and Countability

The Cantor set provides a hands-on example of uncountability. Despite consisting of an infinite number of points, it's considered 'uncountable', much like the set of all real numbers, because there isn't a one-to-one correspondence between the points in the Cantor set and the set of natural numbers. This characteristic delves into the heart of set theory, where different infinities are contrasted and the notion of the size of a set is taken apart and re-defined.

Construction and Decomposition

The creation of the Cantor set is itself a function of set operations, showcasing the removal of subsets (middle thirds) and the formation of a new set by means of infinite union. This recursive pattern of removal and unification is emblematic of the intricate dance between the concepts of whole and part that set theory explores.

Topological Properties

Examining the topological properties of the Cantor set illuminates its unusual nature and offers insight into the breadth and depth of topology as a field of study. At first glance, the Cantor set may appear to have a trivial structure, but upon closer inspection, it reveals a rich topological makeup.By definition, the Cantor set is 'totally disconnected,' meaning it's a set that cannot be broken down into disjoint nonempty open subsets. This property can seem paradoxical since the Cantor set is constructed by literally disconnecting intervals from the unit interval. Nevertheless, these missing intervals form the basis to prove that the Cantor set contains no interval of any positive length within itself. Thus, the Cantor set is non-empty, closed, and devoid of any interior points.The property that the Cantor set is 'perfect' reflects that every point in the Cantor set is a limit point, indicating that for any point you choose, there are other points in the set arbitrarily close by. Lastly, although the set is uncountably infinite, the fact that it can be constructed by removing a countable number of intervals is pivotal to the proof that its complement is a countable union of open intervals.