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Question about rectangles in R^n

  1. Jun 29, 2010 #1
    I know that by definition the cartesian product of [a1, b1], [a2,b2], ..., [an. bn] is a rectangle in R^n. Are we "allowed" to call the cartesian product [a1] * [a2, a2] * ... * [an, an] a rectangle in R^n? I know that by definition [a,b] = {x: a <= x <= b}. The set [a1,a1] = {x: a1 <= x <= a1} which is just the singleton set {a1}. Thus the rectangle [a1] * [a2, a2] * ... * [an, an] is just the cartesian product a1*a2*a3*..*an, where each ai is a real number, which hardly looks like a rectangle if we draw it in say, R^2 or R^3.
  2. jcsd
  3. Jun 29, 2010 #2


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    Science Advisor

    You could call it a degenerate rectangle. It is convenient to extend definitions to degenerate cases, because they occur naturally. In that way you do not need to explicitly mention them whenever you talk about an n-dimensional rectangle.
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