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

1. 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:

My 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

2. 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 $\Bbb R^n$.

Originally Posted by 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 $\Bbb R^n$.

Thanks Euge,

Peter