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  1. MHB Master
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    #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. MHB Master
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    Euge's Avatar
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    #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 $\Bbb R^n$.

  3. MHB Master
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    #3 Thread Author
    Quote Originally Posted by Euge View Post
    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

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