Is the Set of Complex Numbers Open or Closed?Date: 09/20/1999 at 16:27:12 From: Ricardo Subject: The set of complex numbers C: open or closed? In my textbook, _Basic Complex Analysis_, it says that C is open because for each z in C, any epsilon will give abs(w-z) < epsilon for any other complex number w. In other words, it is open because it does not contain its boundary points (because it has no boundary points). To the contrary, when the book later defines a closed set, it states that C is closed because its complement (the null set) is open. In addition, it states that the null set is open because it has no boundary points to test, but it later states that the null set is closed because its complement (C) is open. So each set is both open and closed? ...or neither? Date: 09/20/1999 at 17:03:17 From: Doctor Wilkinson Subject: Re: The set of complex numbers C: open or closed? Hi, Ricardo. The mathematical terms "open" and "closed" are a little confusing, because they sound as if they ought to be opposites, but they're not. A set can easily be neither open nor closed; and it can sometimes be both open and closed. The empty set and the set of all complex numbers are examples of sets that are both open and closed. C is open because if w is any point in C, it is in C, so you don't need the epsilon-condition at all. The empty set is open because it has no points to test. The complement of C is the empty set, and so it's closed (since C is open), and the complement of the empty set is C, and so it's closed also (since the empty set is open). - Doctor Wilkinson, The Math Forum http://mathforum.org/dr.math/ Date: 09/20/1999 at 18:00:37 From: Doctor Schwa Subject: Re: The set of complex numbers C: open or closed? Good thinking. Indeed, they are both open and closed. Open = every point in the set has a neighborhood in the set, so C is open (draw a little circle around any point and it's still in C) and the empty set is open (there aren't any points to draw circles around). Closed = if a sequence of points in the set has a limit, the limit is in the set too. So C is closed (if the limit exists, it's still a complex number) and the empty set is closed (no points in the set, so no limits to worry about). - Doctor Schwa, The Math Forum http://mathforum.org/dr.math/ |
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