A simple corollary of the theorem is that the Cantor set is nonempty, since it is defined as the intersection of a decreasing nested sequence of sets, each of which is defined as the union of a finite number of closed intervals; hence each of these sets is non-empty, closed, and bounded. We use a method of proof known as Cantor’s diagonal argument. urthermore,F as previously discussed, the Cantor set contains no intervals of non-zero length, and so int(C) = ∅. Also, C is bounded since C ⊆ [0,1]. 2,2= (7/9,8/9), and so on. It begins with a discussion of the real algebraic numbers and a statement of his first theorem: The set of real algebraic numbers can be put into one-to-one correspondence with the set of positive integers. The Cantor set is an example of a perfect nowhere dense set, where a perfect set is a closed set with no isolated points and nowhere dense set is a set whose closure has an empty interior. Next, from the two remaining closed intervals we remove the open middle third I. C is compact. In fact, the Cantor set contains uncountably many points. Proof: Each Akis a finite union of closed sets, so Akis closed for all k by Corollary 1(b). The Cantor set is uncountable. Cantor's article is short, less than four and a half pages. Cantor restates this theorem in terms more familiar to mathematicians of his time: The set of real algebraic numbers can be written as an infinite sequence in which each number appears only once. Proof. Since $${\displaystyle {\mathcal {C}}}$$ is a subset of [0,1], its cardinality is also no greater, so the two cardinalities must in fact be equal, by the Cantor–Bernstein–Schröder theorem. It can be shown that there are as many points left behind in this process as there were to begin with, and that therefore, the Cantor set is uncountable. f maps from $${\displaystyle {\mathcal {C}}}$$ onto [0,1]) so that the cardinality of $${\displaystyle {\mathcal {C}}}$$ is no less than that of [0,1]. Then C = ∩Akis also closed by Corollary 1(a). 2,1= (1/9,2/9) and I. Prof.o We have already seen that C is the intersection of closed sets, which implies that C is itself closed. Then C is the Cantor set. Now we will prove some interesting properties of C. 1. 1,1= (1/3,2/3). Suppose instead that C is countable, say C = fx1;x2;x3;x4;:::g. Write x i= 0:d 1 d i 2 d 3 d 4::: as a ternary expansion using only 0s and 2s. The Cantor set is closed and nowhere dense. Also, notice the end points of the intervals at each step are always in the set however, we … Recall C is obtained from the closed interval [0,1] by first removing the open middle third interval I. To see this, we show that there is a function f from the Cantor set $${\displaystyle {\mathcal {C}}}$$ to the closed interval [0,1] that is surjective (i.e. In set theory, Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers.– Such sets are now known as uncountable sets, and the size of infinite sets is now treated …