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Are the following Sets: Open, Closed, Compact, Connected

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  1. Feb 5, 2015 #1
    1. The problem statement, all variables and given/known data
    Ok I created this question to check my thinking.
    Are the following Sets: Open, Closed, Compact, Connected

    Note: Apologies for bad notation.

    S: [0,1)∪(1,2]
    V: [0,1)∩(1,2]

    2. Relevant equations

    S: [0,1)∪(1,2]
    V: [0,1)∩(1,2]

    3. The attempt at a solution

    S: [0,1)∪(1,2]
    Closed - because 0 and 2 represent boundary points
    Compact - because S is closed and bounded
    Not Connected - because it can be separated into two open disjoint sets [0,1) and [2]

    V: [0,1)∩(1,2]
    Open - because the set is (1)
    Not Compact - because it is open
    Connected - because it cannot be separated into two open disjoint sets
     
  2. jcsd
  3. Feb 5, 2015 #2
    Another definition of closed is that the set contains its limit points. I can think of one limit point of S that is not in S. Are you familiar with the definition of a ball or some books call it a neighborhood.

    As for V, as written, V is empty. The ")" means everything up to, but not including.
     
  4. Feb 5, 2015 #3
    Yes I am aware of a ball. It is used in my book to explain the concept of an open set.
    But even so, I thought it was closed, because for the Union I get [0,2] which is, as I understood it, by default closed, given that it includes the boundary points namely, 0 and 2?
    What am I missing? What example do you have?


    As for V . . . didn't realise the intersection there is the null set . . . really thought it was 1 . . . damn.

    Thanks.
     
  5. Feb 5, 2015 #4
    You're getting confused on your interval notation. 1 is not in your set S as written. 1 is not in [0, 1) and 1 is not in (1,0]. But any ball centered at 1 will intersect your set non-trivially, thus 1 is a limit point but it is not in S. So S is not closed.

    Also, closed and not open are not the same thing, a set can be both closed and open or neither open nor closed.
     
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