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Math Help - Problems with Topology question

  1. #1
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    Problems with Topology question

    I'm stuck on a question. It states to consider the set Y = [-1,1] as a subspace of \mathbb{R}, it being the standard topology. And now I have to consider which are open in \mathbb{R} and which are open in Y.

    The first set is A = \{x : 1/2 < |x| < 1\}, which I think is (1/2, 1) \cup (-1, -1/2). Now, that is open in \mathbb{R}, because it is the union of two open sets in \mathbb{R}. But is it open in Y? How do I tell if it is? How about the following cases:

    (1/2, 1] \cup [-1, -1/2)
    [1/2, 1) \cup (-1, -1/2]
    [1/2, 1] \cup [-1, -1/2]

    My guess is that they are all closed in \mathbb{R}, but all open in Y. Am I right? Why/why not?

    Thanks in advance.
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  2. #2
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    Quote Originally Posted by HTale View Post
    I'm stuck on a question. It states to consider the set Y = [-1,1] as a subspace of \mathbb{R}, it being the standard topology. And now I have to consider which are open in \mathbb{R} and which are open in Y.

    The first set is A = \{x : 1/2 < |x| < 1\}, which I think is (1/2, 1) \cup (-1, -1/2). Now, that is open in \mathbb{R}, because it is the union of two open sets in \mathbb{R}. But is it open in Y? How do I tell if it is? How about the following cases:

    (1/2, 1] \cup [-1, -1/2)
    [1/2, 1) \cup (-1, -1/2]
    [1/2, 1] \cup [-1, -1/2]

    My guess is that they are all closed in \mathbb{R}, but all open in Y. Am I right? Why/why not?

    Thanks in advance.
    A subset X is open in [-1,1] iff X = Y \cap [-1,1] where Y is open in \mathbb{R}.

    Now, A = (-1,-1/2) \cup (1/2,1). The set (-1,-1/2) = (-1,-1/2)\cap [-1,1] and (1/2,1) = (1/2,1)\cap [-1,1]. Thus, these are open sets and so the union of two open sets is open.
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  3. #3
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    Quote Originally Posted by HTale View Post
    It states to consider the set Y = [-1,1] as a subspace of \mathbb{R}, it being the standard topology. And now I have to consider which are open in \mathbb{R} and which are open in Y.
    B=(1/2, 1] \cup [-1, -1/2)
    C=[1/2, 1) \cup (-1, -1/2]
    D=[1/2, 1] \cup [-1, -1/2]
    My guess is that they are all closed in \mathbb{R}, but all open in Y. Am I right? Why/why not?
    By “being the standard topology”, I assume you the relative topology on \mathbb{Y}=[-1,1].
    I added letters to your set for reference.

    Set B is neither open nor closed in \mathbb{R}, however it is open in \mathbb{Y}. Can you explain this?


    Set C is neither open nor closed in \mathbb{R}. It is not open in \mathbb{Y} because it contains a boundary point \frac{1}{2} and it is not closed because -1 is a limit point not in the set.

    Set D is closed in \mathbb{R}, and it is closed in \mathbb{Y}.
    Can you explain this?
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  4. #4
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    Quote Originally Posted by Plato View Post
    By “being the standard topology”, I assume you the relative topology on \mathbb{Y}=[-1,1].
    I added letters to your set for reference.

    Set B is neither open nor closed in \mathbb{R}, however it is open in \mathbb{Y}. Can you explain this?


    Set C is neither open nor closed in \mathbb{R}. It is not open in \mathbb{Y} because it contains a boundary point \frac{1}{2} and it is not closed because -1 is a limit point not in the set.

    Set D is closed in \mathbb{R}, and it is closed in \mathbb{Y}.
    Can you explain this?
    Thank you very much, I can explain all of them now.
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