Closed uncountable subsets of the real line

This is post #12 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, II, we show that every subset of the real line that is a perfect set contains a Cantor set and thus has cardinality continuum. We would like to add an observation that every uncountable closed subset of the real line contains a perfect set. Thus every uncountable closed subset of the real line contains a Cantor set and thus has the same cardinality as the real line itself.

Recall that a subset A of the real line is a perfect set if it is closed and every point of A is a limit point of A. In the previous post Perfect sets and Cantor sets, II, we demonstrate a procedure for constructing a Cantor set out of any nonempty perfect set. In another post The Lindelof property of the real line, we show that every uncountable subset A of the real line contains at least one of its limit points. Thus all but countably many points of A are limit points of A. Consequently, for any uncountable closed subset A of the real line, all but countably many points of A are limit points of A. By removing the countably many non-limit points (isolated points), we have 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
9. Perfect sets and Cantor sets, I
10. The Lindelof property of the real line
11. Perfect sets and Cantor sets, II

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One thought on “Closed uncountable subsets of the real line

  1. “By removing the countably many non-limit points (isolated points), we have a perfect set.”

    Isn’t it possible that doing this will not create a perfect set? For example, what if

    A = {1/2, 2/3, 3/4, 4/5, 5/6, …} union {1} union [2,3]?

    Then A is an uncountable closed subset of the real line, but when you throw out all its isolated points, you have “stranded” 1 and you don’t have a perfect set.

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