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I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ...
I am focused on Chapter 1: Continuity ... ...
I need help with an aspect of the proof of Theorem 1.8.17 ... ...
Duistermaat and Kolk's Theorem 1.8.17 and its proof (including the preceding relevant definition) read as follows:https://www.physicsforums.com/attachments/7763
https://www.physicsforums.com/attachments/7764Towards the end of the above proof, D&K write the following:
" ... ... In order to demonstrate that $$K$$ is closed, we prove that $$\mathbb{R}^n \text{\\}K$$ is open. Indeed, choose $$y \notin K$$ and define $$O_j = \{ x \in \mathbb{R}^n \ \mid \ \mid \mid x - y \mid \mid \gt \frac{1}{j} \}$$ for $$j \in \mathbb{N}$$. ... ... "D&K go on to describe the union of the $$O_j$$ as an open cover, implying, of course, the the sets $$O_j$$ are open ... BUT ... why/how are these sets open?
Hope someone can help ...
Peter
I am focused on Chapter 1: Continuity ... ...
I need help with an aspect of the proof of Theorem 1.8.17 ... ...
Duistermaat and Kolk's Theorem 1.8.17 and its proof (including the preceding relevant definition) read as follows:https://www.physicsforums.com/attachments/7763
https://www.physicsforums.com/attachments/7764Towards the end of the above proof, D&K write the following:
" ... ... In order to demonstrate that $$K$$ is closed, we prove that $$\mathbb{R}^n \text{\\}K$$ is open. Indeed, choose $$y \notin K$$ and define $$O_j = \{ x \in \mathbb{R}^n \ \mid \ \mid \mid x - y \mid \mid \gt \frac{1}{j} \}$$ for $$j \in \mathbb{N}$$. ... ... "D&K go on to describe the union of the $$O_j$$ as an open cover, implying, of course, the the sets $$O_j$$ are open ... BUT ... why/how are these sets open?
Hope someone can help ...
Peter
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