2
$\begingroup$

Give an example of an open set in R under the usual topology that is not expressible as the union of a finite number of intervals.

My introduction to proofs class tutorial this week is discussing closed sets... I have feeling giving this example will lead to closed sets?

$\endgroup$
  • 2
    $\begingroup$ Complement of Cantor Set? $\endgroup$ – scitamehtam Apr 12 '15 at 18:05
  • 2
    $\begingroup$ Tangentially related: it is interesting to note that any open set in $\mathbb{R}$ (with the usual topology) must be expressible as the union of a countable number of open intervals. $\endgroup$ – Strants Apr 12 '15 at 20:27
9
$\begingroup$

$$A:= \bigcup_{n \in \mathbb{N}} (n,n+1)$$ isn't because it's an unbounded open set. Any finite union of open intervals equal to it would therefore have to contain an interval of the form $(a, \infty)$, which would contain natural numbers. $A$ doesn't.

$\endgroup$
  • 1
    $\begingroup$ A minor variation with a concise expression is $\Bbb R\setminus\Bbb Z$ (if you extend your union over $\Bbb Z$ instead of $\Bbb N$). $\endgroup$ – Mario Carneiro Apr 13 '15 at 4:08
6
$\begingroup$

Here is a bounded example:

$$[0,1]\setminus\left\{\frac1n:n\in\Bbb Z^+\right\}=\bigcup_{n\in\Bbb Z^+}\left(\frac1{n+1},\frac1n\right)\;.$$

(This is what happens to Frank’s example if you subject it to the function $f(x)=\frac1x$.)

More generally, let $\{(a_n,b_n):n\in\Bbb N\}$ be any countably infinite collection of pairwise disjoint open intervals; then $\bigcup_{n\in\Bbb N}(a_n,b_n)$ is an open set that cannot be expressed as the union of finitely many open intervals.

$\endgroup$
1
$\begingroup$

The first example that occurred to me is the real line itself. Of course this is trivial and uninteresting, never the less it can be helpful to think about simple cases and extend ideas from there.

$\endgroup$
  • $\begingroup$ It is typical to consider infinite or half-infinite intervals like $(\-infty,a]$ to be intervals. In such cases, the real line is not an example. $\endgroup$ – Erick Wong Sep 13 '15 at 16:50

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy