A mistake from Rudin analysis?

In summary: Therefore, in summary, we have discussed the relationship between the closed subset \overline{B_n} and the union of closures \cup_{i=1}^n \overline{A_i}, noting that the former is the smallest closed subset containing B_n and the latter is a closed subset containing B_n. We have also explored the reversed inclusion \overline{B_n}\supset \cup_{i=1}^n \overline{A_i}, which holds for finite unions but not necessarily for infinite unions. An example was provided to illustrate this concept.
  • #1
jessicaw
56
0
Let [tex] B_n=\cup_{i=1}^n A_i [/tex].
[tex] \overline{B_n}[/tex] is the smallest closed subset containing [tex] B_n[/tex].
Note that
[tex]\cup_{i=1}^n \overline{A_i}[/tex] is a closed subset containing [tex] B_n[/tex].
Thus,
[tex] \overline{B_n}\supset \cup_{i=1}^n \overline{A_i}[/tex]Isn't the truth should be that
[tex] \overline{B_n}[/tex] is the smallest?
How come claim that
[tex]\cup_{i=1}^n \overline{A_i}[/tex] is even smaller?
 
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  • #2
I agree; the fact that the union of closures is a closed subset containing B_n, combined with minimality of cl(B_n), gives
[tex] \overline{B_n}\subset \cup_{i=1}^n \overline{A_i}.[/tex]
In fact the reversed inclusion
[tex] \overline{B_n}\supset \cup_{i=1}^n \overline{A_i}[/tex]
also holds if the union is finite (i.e. the closure operation distributes over finite unions), but not if the union is infinite. But that requires a different argument, so I don't know what Rudin is doing (I don't have his book).
 
  • #3
For example, we could argue as follows:
[tex]A_i\subseteq \overline{A_i}\ \forall i[/tex]

[tex]\Rightarrow \bigcup_i A_i\subseteq \bigcup_i \overline{A_i}[/tex]

[tex]\Rightarrow \overline{\bigcup_i A_i}\subseteq \overline{\bigcup_i \overline{A_i}}[/tex]

A finite union of closed sets is closed, so if I is finite then

[tex]\overline{\bigcup_i \overline{A_i}}=\bigcup_i \overline{A_i}[/tex]

which proves the reversed inclusion

[tex]\overline{\bigcup_i A_i}\subseteq \bigcup_i \overline{A_i}.[/tex]

However, an infinite union of closed sets is not necessarily closed, making this argument stop working. Indeed, consider

[tex]I=\mathbb{Q},\ A_q=\{q\}.[/tex]

Then

[tex]\overline{A_q}=\overline{\{q\}}=\{q\}[/tex].

Hence

[tex]\overline{\bigcup_{q\in I}A_q}=\overline{\mathbb{Q}}=\mathbb{R}[/tex]

[tex]\bigcup_{q\in I}\overline{\{q\}}=\mathbb{Q}.[/tex]
 

Related to A mistake from Rudin analysis?

1. What is "A mistake from Rudin analysis"?

"A mistake from Rudin analysis" refers to a mathematical error or inconsistency found in Walter Rudin's book "Principles of Mathematical Analysis". This book is a well-known and highly regarded textbook in the field of real analysis, but it has been found to contain a mistake in one of its proofs.

2. What was the mistake found in Rudin's analysis?

The mistake in Rudin's analysis was found in the proof of Theorem 2.30, which deals with the convergence of sequences in a metric space. It involves the use of a lemma that was proven incorrectly, leading to an incorrect conclusion in the theorem's proof.

3. How was the mistake in Rudin's analysis discovered?

The mistake was first discovered by a mathematician named Terence Tao, who posted about it on his blog in 2007. Other mathematicians then read and verified Tao's findings, and the mistake was confirmed. Since then, the mistake has been discussed and studied by many mathematicians.

4. Has the mistake in Rudin's analysis been corrected?

Yes, the mistake has been corrected. After the mistake was discovered, Walter Rudin himself admitted the error and published a corrected version of the proof in the 3rd edition of his book. The corrected proof can also be found in other analysis textbooks, such as "Real Analysis" by Royden and Fitzpatrick.

5. How does the mistake in Rudin's analysis impact the field of mathematics?

The mistake in Rudin's analysis does not significantly impact the field of mathematics as a whole. It serves as a reminder that even the most respected and well-known mathematicians can make mistakes, and that the process of finding and correcting errors is an essential part of mathematical research. The mistake also highlights the importance of peer review and communication among mathematicians.

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