The Cantor set, III

This is post # 8 of the series on the Euclidean topology of the real line. See the links at the bottom for other posts in the series.

The Cantor set, I
The Cantor set, II

In two previous posts (see links above), we discuss the “middle third” Cantor set constructed on the unit interval. In this post, we discuss two variations on this construction. In the first variation, the lengths of the removed intervals sum to 1 (hence these Cantor sets have zero Lebesgue measure). The middle third construction is a specific example if this algorithm. In the second variation, the sum of the lengths of removed intervals is less than one. Thus these Cantor sets have positive Lebesgue measure. However, the resulting Cantor sets from either approach have the same topological properties.

The Middle k Cantor Set
Let 0<k<1. In this construction, we remove the middle k^{th} open interval of [0,1]. This removed open interval has length k and the remaining set has length 1-k.

In the second stage, for each of the two remaining two closed intervals, we remove the middle k^{th} open interval. The two removed open intervals have a combined length of (1-k)k. Thus, the length of the remaining set is (1-k)-(1-k)k=(1-k)^2.

In the third stage, the four removed intervals have a combined length of (1-k)^2k. Then the remaining set has length:

(1-k)^2-(1-k)^2k=(1-k)^3.

When the inductive process is completed, the set of the remaining points is a Cantor set. The total length of removed open intervals is:

\displaystyle k+(1-k)k+(1-k)^2k+ \cdots=1

When k=\frac{1}{3}, we have the middle third Cantor set that is constructed in the previous post The Cantor set, I. Whether k=\frac{1}{3} or another fraction, the resulting Cantor sets have the same properties. This middle k process always produces a compact subset of the real line that has cardinality continuum, which is also a perfect set and a nowhere dense set in [0,1] as well as a totally disconnected set. In other words, all the properties discussed in the two previous posts (see the links indicated above) hold for the Cantor sets produced by this middle k algorithm.

The Cantor Sets of Positive Measure
Pick a sequence \left\{a_n\right\} such that 0<\sum \limits_{n=1}^{\infty} a_n<1. For the illustration here, we use a_n=\frac{1}{2^{n+1}} where n=1,2,3,\cdots. Note that \sum \limits_{n=1}^{\infty} a_n=\frac{1}{2}.

In this construction, we still remove the middle open interval in any remaining closed interval. The key is that in each stage we remove open intervals with a total length of a_n=\frac{1}{2^{n+1}}. In the end, the removed intervals have a total length of \frac{1}{2}, making the left over piece of length \frac{1}{2}.

In stage n=1, we remove the middle \frac{1}{2^2} open interval of [0,1]. In stage n=2, in each of the remaining two closed intervals, we remove the middle part such that the two removed open intervals have equal length and the combined length of \frac{1}{2^3}.

In stage n=3, the four removed open intervals are of equal length and with a combined total length of \frac{1}{2^4}.

Continue with the inductive process, making sure that any removed open interval is the middle part of the remaining closed interval. When it is done, the removed open intervals have a total length of \frac{1}{2}. The set of the remaining points is a Cantor set of length \frac{1}{2}. Other than having positive Lebesgue measure, the Cantor sets produced by this algorithm have the same topological properties as the middle third Cantor set, namely, being totally disconnected and a perfect set.

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
The Cantor set, I
The Cantor set, II

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