Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I was trying to prove something, and I did it, but what I used is too exaggerated. The problem is:

Let K be the cantor set, prove that the sets $$ \eqalign{ & \left\{ {\left| {x - y} \right|\,:x,y \in K} \right\} \cr & \left\{ {x + y\,\,:\,x,y \in K} \right\} \cr} $$ are closed in the real numbers.

What I did is said that these functions are continuous, and because $K \times K$ is compact then its image is also compact, and so the set must be closed in $\mathbb{R}$. Can I prove this directly?

share|cite|improve this question
Your proof is a beautiful one-liner. Why would you want something else? A "more direct" proof will likely just explicitly reprove the facts you've used. – Anton Geraschenko Aug 23 '11 at 18:18
Like Anton said. Or, prove that the first set is $[0,1]$ and the second one is $[0,2]$. – Did Aug 23 '11 at 19:25
How is using basic facts of point-set topology too exaggerated? – JSchlather Aug 23 '11 at 19:53

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.