Infinite intersection of open sets in C that is closed

autre
Messages
116
Reaction score
0

Homework Statement



Find an infinite intersection of open sets in C that is closed.

The Attempt at a Solution



Consider the sets A_n = (-1/n,1/n). Since 0 in A_n for all n, 0 in \bigcap A_{n}. Here I'm a little stuck -- is the proof in R analogous to the proof in C, or do I need a different example?
 
Physics news on Phys.org
I can't discern whether the example you give for ℝ is drawn from the book, or whether you're attempting to construct the proof first for ℝ and then generalize to ℂ. If it's the former then skip to the second paragraph, if it's the latter then try assuming there was another point in the infinite intersection and see if you can't derive a contradiction.

Your open interval (-1/n,1/n) in ℝ can more generally be called an open ball in ℝ centered at zero. There's a reason they call them open balls, think about it in ℂ.
 
Last edited:
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...

Similar threads

Back
Top