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Thread: Closed and open set book example

  1. #1
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    Closed and open set book example

    Consider the following subset of the real line:
    $\displaystyle Y=[0,1]\cup (2,3)$,
    in the subspace topology. In this space, the set [0,1] is open, since it is the intersection of an open set in R with Y. Similar (2,3) is open as a subset of Y; it is even open as a subset of R (I understand everything here). Since [0,1] and (2,3) are complements in Y of each other, we conclude that both [0,1] and (2,3) are closed as subsets of Y.

    Y-[0,1] is open so [0,1] is closed.
    Y-(2,3) is closed so why is (2,3) closed?
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    MHF Contributor Drexel28's Avatar
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    Re: Closed and open set book example

    Quote Originally Posted by dwsmith View Post
    Consider the following subset of the real line:
    $\displaystyle Y=[0,1]\cup (2,3)$,
    in the subspace topology. In this space, the set [0,1] is open, since it is the intersection of an open set in R with Y. Similar (2,3) is open as a subset of Y; it is even open as a subset of R (I understand everything here). Since [0,1] and (2,3) are complements in Y of each other, we conclude that both [0,1] and (2,3) are closed as subsets of Y.

    Y-[0,1] is open so [0,1] is closed.
    Y-(2,3) is closed so why is (2,3) closed?
    The idea is this, let $\displaystyle [0,1]\subseteq Y$ is equal to $\displaystyle [0,1]\cap Y$ when $\displaystyle [0,1]$ is thought of as a subset of $\displaystyle \mathbb{R}$ and so closed in $\displaystyle Y$ with the subspace topology (being the intersection of $\displaystyle Y$ with a set closed in $\displaystyle \mathbb{R}$). Using the exact same idea you can show that $\displaystyle (2,3)$ is open in $\displaystyle Y$. But, let me ask you this, what is $\displaystyle Y\cap (-\infty,1.5)$?
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    Re: Closed and open set book example

    Quote Originally Posted by Drexel28 View Post
    The idea is this, let $\displaystyle [0,1]\subseteq Y$ is equal to $\displaystyle [0,1]\cap Y$ when $\displaystyle [0,1]$ is thought of as a subset of $\displaystyle \mathbb{R}$ and so closed in $\displaystyle Y$ with the subspace topology (being the intersection of $\displaystyle Y$ with a set closed in $\displaystyle \mathbb{R}$). Using the exact same idea you can show that $\displaystyle (2,3)$ is open in $\displaystyle Y$. But, let me ask you this, what is $\displaystyle Y\cap (-\infty,1.5)$?
    I understand why [0,1] is open and closed. I don't understand why (2,3) is closed though.
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Re: Closed and open set book example

    Quote Originally Posted by dwsmith View Post
    I understand why [0,1] is open and closed. I don't understand why (2,3) is closed though.
    Either because $\displaystyle (2,3)=Y-[0,1]$ or because $\displaystyle [2,3]=[1.5,10]\cap Y$.
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