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Find the Closure of these subsets

  1. Sep 19, 2007 #1
    1. The problem statement, all variables and given/known data
    X=R real numbers, U in T, the topology <=> U is a subset of R and for each s in U there is a t>s such that [s,t) is a subset of U where [s,t) = {x in R; s<=x<t}

    Find the closure of each of the subsets of X:

    (a,b), [a,b), (a,b], [a,b]



    3. The attempt at a solution
    I don't understand the topology of X very well. So have trouble finding the closure. No metric is allowed?

    I used complements to work out open and closed properties and came to the conclusion :
    [a,b), [a,b), [a,b] and [a,b] as the closures of the above respectively.
     
    Last edited: Sep 19, 2007
  2. jcsd
  3. Sep 19, 2007 #2

    matt grime

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    You seem to have an obsession with metrics. Stop it. Firstly, try to work out what a typical open set looks like, thus you know what a typical closed set looks like. For instance, is (0,1) open? Is it closed? (It can of course be both.) What about (-inf,0], (-inf,0), (0,inf), or [0,inf)?

    And what's stopping you making a sensible guess? I mean it is surely the case that the closure of (a,b) is going to be one of (a,b), (a,b], [a,b) or [a,b], so you have to see which of those is closed, i.e. which has open complement.
     
    Last edited: Sep 19, 2007
  4. Sep 19, 2007 #3
    My guesses are displayed in the OP.
     
  5. Sep 20, 2007 #4

    matt grime

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    And what are your justifications for them?
     
  6. Sep 20, 2007 #5

    HallsofIvy

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    Do you remember back in Calculus I (or maybe Precalculus) when you worked with "closed intervals" and "open intervals"? Those names come from this. Is (a, b) a closed or open interval? What can you do to make it closed?
     
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