# analysis help (countable union of disjoint sets)

• Sep 30th 2010, 04:51 PM
DontKnoMaff
analysis help (countable union of disjoint sets)
Show that if \$\displaystyle E \subseteq R\$ is open, then \$\displaystyle E \$ can be written as an at most countable union of disjoint intervals, i.e., \$\displaystyle E=\bigcup_n(a_n,b_n)\$. (It's possible that \$\displaystyle a_n=-\inf\$ or \$\displaystyle b_n=+\inf\$ for some \$\displaystyle n\$.) Hint: One way to do this is to put open intervals around each rational point in E in such a way that every point of E and only points of E are contained somewhere in these intervals. Then combine the intervals that intersect.

OK, intuitively, I get this, but what confuses me about the method they suggest is when I'm told to combine the intervals that intersect. Doesn't that imply that they aren't disjoint?! At firstI had the idea to take the set of all Neigborhoods of all the rationals in E of rational radius that do not intersect the complement of E, but these are not disjoint. Then I decided to take an arbitrarily large neighborhood in E and then take neighborhoods of the space that is left over and keep filling in the gaps with more and more neighborhoods until I have an at most countable amount of neighborhoods that are dense in E, but I had trouble getting that down and I also figured I should probably utilize the hint... But the hint confuses me more than the problem statement.
• Sep 30th 2010, 05:40 PM
Iondor
The hint means, that the "union" of open intervals that are not disjoint is again an open interval. So, if you have any countable union of not necessarily disjoint open intervals, you can make it into a countable union of disjoint open intervals by combining each cluster of intersecting intervals into one interval.

Alternatively you can take for any rational point x in E the "largest" open interval containing x that is still a subset of E.
This is the same as the union of all open intervals containing x, which are subsets of E.
Then show that two such largest intervals must either be disjoint or equal and that their union covers E.