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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/   
    
Associated Topics:
High School Imaginary/Complex Numbers
High School Sets

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