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Constructing a Set with the following characteristics

  1. May 19, 2009 #1
    1. The problem statement, all variables and given/known data

    Construct a set ScR such that S, the interior of S, the closure of S, the closure of the interior of S, and the interior of the closure of S are all distinct (ie no 2 of them are equal)


    2. Relevant equations
    closure of S - smallest closed set containing S
    interior of S - set of all points in S for which S is a neighbourhood



    3. The attempt at a solution

    I am really having trouble starting this...or more precisely, i'm having trouble seeing how this question is possible at all...How can a set, its interior, and its closure all be distinct? If a set is open, then its interior is simply equal to the set itself, so that leaves me with only closed sets to consider. But, if a set is closed, then it contains all of its boundary points, so the closure of S is equal to S...
    so the only exception i could think of is the open/closed sets...

    but so far the closest ive gotten to something that satisfies all of those things is the set of rational numbers
    S = Q
    int S = empty set
    Closure of S = Real number line
    but then closure of the interior of S = closure of the empty set = empty set
    so thats equal to int S and doesn't work
    and the interior of the closure of S = real number line = closure of S so that doesn't work either...help!
     
  2. jcsd
  3. May 19, 2009 #2
    I just finished answering this question from someone else. You can find the thread here:
    <https://www.physicsforums.com/showthread.php?t=314690>
    Enjoy!
     
  4. May 19, 2009 #3
    Oh wow...awesome! thanks :) (its probably a classmate of mine haha...since both of us thought of rational numbers, and thats something we discussed in class)
     
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