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Math Help - A question on Open and Closed Sets that I don't understand

  1. #1
    Member akhayoon's Avatar
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    A question on Open and Closed Sets that I don't understand

    let  \phi be the empty set. Let X be the real line, the entire plane, or, in the three dimensional case, all of three-space. For each subset of A of X, let
    X - A be the set of all points x  \in X such that x NOT  \in A

    (a) show that \phiis both open and closed

    (b) show the X is both open and closed (it can be shown that  \phi and X are the only subsets of X that are both open and closed)

    (c) let U be a subset of X. Show that U is open iff X - U is closed

    (d) Let F be a subset of X. Show that F is closed iff X - F is open.


    I understand the concept of open and closed sets, but the thing is I don't understand what the question is describing to me before any of parts a,b,c or d in the first place. If I got a little bit on that I could actually understand how to solve the problem.
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  2. #2
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    Open set - Wikipedia, the free encyclopedia
    Closed set - Wikipedia, the free encyclopedia
    http://en.wikipedia.org/wiki/Empty_set

    A subset U of the Euclidean n-space Rn is called open if, given any point x in U, there exists a real number ε > 0 such that, given any point y in Rn whose Euclidean distance from x is smaller than ε, y also belongs to U. Equivalently, U is open if every point in U has a neighbourhood contained in U.
    In Euclidean spaces, a set A \subseteq \mathbb{R}^n is closed if the limit of every convergent sequence of points in A, is in A itself.

    What you need to do in the question:

    A) Show that \phi is both open and closed.
    B)  \mathbb{R}, \mathbb{R}^2, \mathbb{R}^3 are both open and closed. Try doing this only for \mathbb{R}, the rest are pretty much the same.
    C) and D) are both done with proofs by contradiction. Can you think how?
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  3. #3
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    Quote Originally Posted by akhayoon View Post
    let [tex]
    I understand the concept of open and closed sets.
    Well there are many different definitions that apply here.
    What definition of open set were you given?
    What definition of closed set were you given?
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