Question about rectangles in R^n

  • Thread starter JG89
  • Start date
  • #1
726
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.
 

Answers and Replies

  • #2
disregardthat
Science Advisor
1,866
34
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.
 

Related Threads on Question about rectangles in R^n

Replies
1
Views
1K
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
1
Views
5K
Replies
11
Views
1K
  • Last Post
Replies
2
Views
4K
  • Last Post
Replies
15
Views
2K
  • Last Post
Replies
1
Views
5K
  • Last Post
Replies
23
Views
4K
  • Last Post
Replies
4
Views
2K
Replies
17
Views
3K
Top