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

[Math Amateur]

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 $$\mathbb{R}^n$$ ... 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

 

[Euge]

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 $\Bbb R^n$.

 

[Math Amateur]

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