# Infinite intersection of open sets in C that is closed

1. Jan 18, 2012

### autre

1. The problem statement, all variables and given/known data

Find an infinite intersection of open sets in C that is closed.
3. 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?

2. Jan 18, 2012

### Poopsilon

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: Jan 18, 2012