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    I have the following question and i'm not sure how to go about proving whether sets are open closed, or both.


    Which of the following sets are open and which are closed?


     1)\ [1,2] \cup [3,4] \ in \ ( \mathbb{R},|\cdot|)


     2)[1.2] \ in (\mathbb{R},d) where d is a discrete metric.


     3)\ B=[x=(x_n)_(n\in\mathbb{N}) :|x_n|< \epsilon\ for  \ all \ n \in \mathbb{N}] where \epsilon>0 is given in (\ell^\infty, ||\cdot||_\infty)


    4)\ c_0 the set of sequences converging to 0 in (\ell^\infty,||\cdot||~_\infty)


    For question 1 i have used the proof that a finite union of closed sets is closed, and hence it is closed.. which i think is right... but for the others any help would be greatly appreciated
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    Just write down a basis for the topology of each and it becomes obvious.

    I mean, do you know what a discrete metric is? I can't imagine so if you don't know whether that set is open or closed or otherwise. Look up the definitions of the metrics you are using.
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    (Original post by Mark85)
    Just write down a basis for the topology of each and it becomes obvious.

    I mean, do you know what a discrete metric is? I can't imagine so if you don't know whether that set is open or closed or otherwise. Look up the definitions of the metrics you are using.

    I didn't mean to post the discrete metric question as I know this is both open and closed....


    but I am really unsure of how to answer parts 3 and 4.. Especially 4..-any help in pointing me in the right direction would be great!
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    (Original post by gemma331)
    I didn't mean to post the discrete metric question as I know this is both open and closed....


    but I am really unsure of how to answer parts 3 and 4.. Especially 4..-any help in pointing me in the right direction would be great!
    What are you having trouble with in particular?

    Do you know what it means for a set to be closed or open in each of those spaces? Can you visualise what the set of 3 and 4 look like?
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    (Original post by Mark13)
    What are you having trouble with in particular?

    Do you know what it means for a set to be closed or open in each of those spaces? Can you visualise what the set of 3 and 4 look like?

    I'm completely lost and don't know where to even start with either of them!
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    (Original post by gemma331)
    I'm completely lost and don't know where to even start with either of them!
    Ok, let's have a look at 3.

    To check whether B is closed, you need to take an arbitrary sequence of points in B converging to a limit, and check to see if that limit must be in B - if the limit must be in B, then B is closed. To check if B is open, you need to see whether every point of B is contained in an open set which itself is entirely contained in B. Does that make sense? There are other ways of looking at openness/closedness but these are fine for now.
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    (Original post by Mark13)
    Ok, let's have a look at 3.

    To check whether B is closed, you need to take an arbitrary sequence of points in B converging to a limit, and check to see if that limit must be in B - if the limit must be in B, then B is closed. To check if B is open, you need to see whether every point of B is contained in an open set which itself is entirely contained in B. Does that make sense? There are other ways of looking at openness/closedness but these are fine for now.
    yeah I understand that, but I'm struggling to know which sequence to pick etc
 
 
 
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