This is post # 6 of the series on the Euclidean topology of the real line. See the links at the bottom for other posts in the series.
The goal is this post is to construct the Cantor set and demonstrate that the Cantor set has the same cardinality as the set of all real numbers. In the next post we discuss the topological properties of the Cantor set.
For each positive integer , let be the set of all sequences zeros and ones of length . Each has sequences. For each , we can tag on a or and obtain an element of . For example, for , both .
Let be the unit interval. Remove the middle third open interval from . Label the remaining two intervals by and . Let be the union of these two closed intervals.
Consider the stage at . For , remove the middle third of and obtain and . For , remove the meddle third of and obtain and . Let . Note that the length of each closed interval is .
Suppose we have the following at the stage, where .
- The closed interval has been defined for each .
- The length of each is .
We now define the closed intervals in stage . For each closed interval , we remove the middle third open intervals and obtain two closed intervals and where each has length . Let be defined as:
We now define the Cantor set . This is a nonempty set subset of the unit interval. The point since . Being the union of finitely many closed intervals, each is a closed subset of . Thus each is a compact subset of . Any nested decreasing sequence of compact subsets of the real line is nonempty. See the post The Euclidean topology of the real line (4) – Compactness. In the remainder of this post, we show that the Cantor set has the cardinality continuum.
Let and be sets. A function is a one-to-one function if for with , , that is, distinct elements of the domain are mapped to distinct elements of the range. A function is said to be a surjection if i, i.e. every point in the set is mapped by some point of . A function is said to be a bijection if it is both a one-to-one function and a surjection.
The sets and have the same cardinality if there is a bijection . When and have the same cardinality, we use the notation . For example, the set of all rational numbers has the same cardinality as the integers since we can define a bijection between the two sets. More informally, we can put the rational numbers in an exhaustive list indexed by the integers. When a set has the same cadinality as the set of real numbers, we say that the set has cardinality continuum.
When there is a one-to-one function , we denote this condition by . Obviously when , we have . The Cantor–Bernstein–Schroeder theorem states that when there are one-to-one functions and , we have . In other words, if and , we have . We utilize this fact when we show that the Cantor set and the set of all real numbers have the same cardinality.
The Cardinality of the Cantor Set
Let be the set of all nonnegative integers. Let be the set of all infinite sequences of 0’s and 1’s. Let be the set of subsets of . The cardinality of the Cantor set follows from the following relationship:
The inequality is clear since the Cantor set is a subset of the real line. The following claims establish the relationship.
This follows from the contruction of the Cantor set. Let . At each stage of the construction, there are many closed intervals and the point belongs to exactly one of these intervals, say . At the next stage, the point belongs to one of the closed intervals as a result of splitting by removing the middle third. Since the lengths of approaches to zero, we have .
Note that for each and agree on the first coordinates. Thus the finite sequences form an countably infinite sequence such that and agree on the first coordinates of . Let .
We just describe a mapping . This mapping is one-to-one. For where , there must be some stage where and belong to disjoint closed intervals. Otherwise, . Let be the least integer such that and belong to disjoint closed intervals. Then and belong to the same interval in stage . However, and belong to disjoint closed intervals at stage , say and . As a result, and disagree at the coordinate, implying .
No point in is missed by the mapping . For each , let be the sequence of its first coordinates. Then the sequence of closed intervals collapses to a point . It is clear that .
The surjection is defined by an indicator function. For each , such that the coordinate is:
It can be verified that is a one-to-one map and that .
Let be the set of all rational numbers. Since , we have . We now define a one-to-one map .
For each , let . It is clear that for .
Links to previous posts on the topology of the real line:
The Euclidean topology of the real line (1)
The Euclidean topology of the real line (2)
The Euclidean topology of the real line (3) – Completeness
The Euclidean topology of the real line (4) – Compactness
The Cantor bus tour