Open Subsets of R^n .... D&K Lemma 1.2.5

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In summary, Duistermaat and Kolk's proof of Lemma 1.2.5 (ii) in Chapter 1 of "Multidimensional Real Analysis I: Differentiation" is only stated and proved for a finite collection of open subsets of $\mathbb{R}^n$. This is because the infinite collection case presents difficulties, as shown by the example of the intersection of open cubes in $\mathbb{R}^n$. This raises the question of why the restriction to finite collections, but the authors do not provide an explanation for this.
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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 Lemma 1.2.5 (ii) ...

Duistermaat and Kolk"s statement and proof of Lemma 1.2.5 reads as follows: View attachment 7673My question regarding Lemma 1.2.5 is as follows:

Lemma 1.2.5 (ii) is stated and proved only for a finite collection of open subsets of $$\displaystyle \mathbb{R}^n$$ ... but why do we restrict the result to finite collections of open subsets ... there must be a problem with the infinite collection case ... but D&K give no explanation of why this is so ...

Can someone please explain the difficulty with the infinite collection case ...

Hope someone can help ...

Peter

The intersection of open cubes $$C_m := \left(-\frac{1}{m}, \frac{1}{m}\right)\times\cdots \times\left(-\frac{1}{m}, \frac{1}{m}\right)\quad (m = 1,2,3,\ldots)$$ in $\Bbb R^n$ is the set containing only the origin $\bf 0$, but $\{\bf 0\}$ is not open in $\Bbb R^n$.

Euge said:
The intersection of open cubes $$C_m := \left(-\frac{1}{m}, \frac{1}{m}\right)\times\cdots \times\left(-\frac{1}{m}, \frac{1}{m}\right)\quad (m = 1,2,3,\ldots)$$ in $\Bbb R^n$ is the set containing only the origin $\bf 0$, but $\{\bf 0\}$ is not open in $\Bbb R^n$.
Thanks Euge,

Peter

What are open subsets of R^n?

Open subsets of R^n refer to sets that contain all points within a specific region of n-dimensional space, where the boundaries of the region are not included in the set. In other words, the set does not include its boundary points.

What is D&K Lemma 1.2.5?

D&K Lemma 1.2.5, also known as the Dehn and Kazhdan Lemma, is a mathematical theorem that states that for any open subset in R^n, there exists a compact subset within the set that has the same homotopy type.

How is D&K Lemma 1.2.5 used in mathematics?

D&K Lemma 1.2.5 is used in mathematics to prove certain topological properties of open subsets in R^n. It is also used in the study of geometric topology and algebraic topology.

What is the significance of open subsets in mathematics?

Open subsets are important in mathematics because they allow for the analysis and study of continuous functions and topological spaces. They also play a crucial role in the development of calculus and differential equations.

Can open subsets of R^n be used in other fields of study?

Yes, open subsets of R^n have applications in various fields such as physics, engineering, and computer science. They are used to model and analyze systems that have continuous properties and require a topological understanding.

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