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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 the proof of Lemma 1.2.10 ...

    Duistermaat and Kolk"s proof of Lemma 1.2.10 (including D&K's definition of a cluster point and the closure of a set) reads as follows:





    In the above proof of Lemma 1.2.10 we read the following:

    "... ... Thus ( $ \displaystyle \overline{A} )^c = \text{int(}A^c)$, or $ \displaystyle \overline{A} = \text{(int(} A^c))^c$, which implies that $ \displaystyle \overline{A}$ is closed in $ \displaystyle \mathbb{R}^n$. ... ...


    Can someone please explain (preferably in detail) how/why

    $ \displaystyle \overline{A} = \text{(int(} A^c))^c$

    implies that

    $ \displaystyle \overline{A}$ is closed in $ \displaystyle \mathbb{R}^n$. ... ...



    Help will be much appreciated ... ...

    Peter


    ============================================================================


    It may be helpful for MHB members reading the above post to have access to D&K's definition of an open set ... so I am providing the same ... as follows ... :





    ... and a closed set is simply a set whose complement is open ... ...

    Hope that helps ...

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    #2
    Quote Originally Posted by Peter View Post
    Can someone please explain (preferably in detail) how/why

    $ \displaystyle \overline{A} = \text{(int(} A^c))^c$

    implies that

    $ \displaystyle \overline{A}$ is closed in $ \displaystyle \mathbb{R}^n$.
    The interior of a set is open. So $\operatorname{int}(A^c)$ is an open set and therefore its complement $(\operatorname{int}(A^c))^c$ is a closed set.

  3. MHB Craftsman
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    #3
    Quote Originally Posted by Opalg View Post
    The interior of a set is open. So $\operatorname{int}(A^c)$ is an open set and therefore its complement $(\operatorname{int}(A^c))^c$ is a closed set.
    The whole point is to prove that the interior of a set is open. This does not follow directly from definition 1.2.2 above; it may be (and should be) proved somewhere else in the book.

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