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