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I am reading Walter Rudin's book, Principles of Mathematical Analysis.

Currently I am studying Chapter 2:"Basic Topology".

I am concerned that I do not fully understand the proof of Theorem 2.43 concerning the uncountability of perfect sets in \(\displaystyle R^k\).

Rudin, Theorem 2.43 reads as follows:

View attachment 3806

In the above proof, Rudin writes:

"Let \(\displaystyle V_1\) be any neighbourhood of \(\displaystyle x_1\). If \(\displaystyle V_1\) consists of all \(\displaystyle y \in R^k\) such that \(\displaystyle | y - x_1 | \lt r\), the closure \(\displaystyle \overline{V_1}\) of \(\displaystyle V_1\) is the set of all \(\displaystyle y \in R^k\) such that \(\displaystyle | y - x_1 | \le r\)."

Now, I am assuming that the above two sentences that I have quoted from Rudin's proof are a recipe or formula to be followed in constructing \(\displaystyle V_2, V_3, V_4\), and so on ... ... is that right?

I will assume that is the case and proceed ...

Rudin, then writes:

"Suppose \(\displaystyle V_n\) has been constructed, so that \(\displaystyle V_n \cap P\) is not empty ... ... "My question is as follows:

Why does Rudin explicitly mention that he requires \(\displaystyle V_n\) to be constructed so that \(\displaystyle V_n \cap P\) is not empty?

Surely if \(\displaystyle V_n\) is constructed in just the same way as \(\displaystyle V_1\) then \(\displaystyle V_n\) is a neighbourhood of \(\displaystyle x_n\) ... ... and therefore we are assured that \(\displaystyle V_n \cap P\) is not empty ... ... aren't we? ... ... and so there is no need to mention that \(\displaystyle V_n\) needs to be constructed in a way to assure this ...

Can someone please clarify this issue ...

Hope someone can help ...

Peter

Currently I am studying Chapter 2:"Basic Topology".

I am concerned that I do not fully understand the proof of Theorem 2.43 concerning the uncountability of perfect sets in \(\displaystyle R^k\).

Rudin, Theorem 2.43 reads as follows:

View attachment 3806

In the above proof, Rudin writes:

"Let \(\displaystyle V_1\) be any neighbourhood of \(\displaystyle x_1\). If \(\displaystyle V_1\) consists of all \(\displaystyle y \in R^k\) such that \(\displaystyle | y - x_1 | \lt r\), the closure \(\displaystyle \overline{V_1}\) of \(\displaystyle V_1\) is the set of all \(\displaystyle y \in R^k\) such that \(\displaystyle | y - x_1 | \le r\)."

Now, I am assuming that the above two sentences that I have quoted from Rudin's proof are a recipe or formula to be followed in constructing \(\displaystyle V_2, V_3, V_4\), and so on ... ... is that right?

I will assume that is the case and proceed ...

Rudin, then writes:

"Suppose \(\displaystyle V_n\) has been constructed, so that \(\displaystyle V_n \cap P\) is not empty ... ... "My question is as follows:

Why does Rudin explicitly mention that he requires \(\displaystyle V_n\) to be constructed so that \(\displaystyle V_n \cap P\) is not empty?

Surely if \(\displaystyle V_n\) is constructed in just the same way as \(\displaystyle V_1\) then \(\displaystyle V_n\) is a neighbourhood of \(\displaystyle x_n\) ... ... and therefore we are assured that \(\displaystyle V_n \cap P\) is not empty ... ... aren't we? ... ... and so there is no need to mention that \(\displaystyle V_n\) needs to be constructed in a way to assure this ...

Can someone please clarify this issue ...

Hope someone can help ...

Peter

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