1

Mathematical analysis by Tom Apostol

Definition. let $S$ be an open subset of $\Bbb R$. An open interval $I$ (which may be finite or infinite) is called component interval of $S$ if $I \subset S$ and if there is no open interval $J \ne I$ such that $I \subseteq J$ and $J \subseteq S$.

Now, I can't understand what it says. If I take $S =(1,2)$ , then is $(1,2)$ its only component interval?

Then how can $\Bbb R$ be a union of such open intervals?

Examples will be very helpful.

Hosein Rahnama
  • 14,882
divya
  • 95
  • $\mathbb R=(-\infty,\infty)$ is also an open interval. (The definition is more interesting for disconnected open sets, like $(0,1)\cup(2,3)$.) – Akiva Weinberger Aug 18 '15 at 17:59
  • Yes, if $S = (1,2)$ then it has exactly one component interval, namely $(1,2)$. It is therefore the union of this one interval. –  Aug 18 '15 at 17:59
  • Can you come up with an open set with infinitely many component intervals, by the way? – Akiva Weinberger Aug 18 '15 at 18:01
  • Regarding your last question, could it be that the statement is rather: Any open subset of $\mathbb R$ is the disjoint uinon of countabl ymany component intervals? – Hagen von Eitzen Aug 18 '15 at 18:06
  • Note that the term "countable" is slightly ambiguous, since some use it to mean "countably infinite" and others use it to mean "at most countable" (i.e. possibly finite). Hopefully, your book is consistent with itself, though it's usually clear from context. – Akiva Weinberger Aug 18 '15 at 18:09
  • @columbus8myhw union of intervals (n, n+1) , n is positive integer, is an open set with countably infinite component intervals. But this is very obvious example. Can I get few good examples? I can't think of any. – divya Aug 18 '15 at 19:43
  • 1
    @divya How about $(.1,1)\cup(.01,.1)\cup(.001,.01)\cup\dotsb$? Or the union of every other set in that. Or, for a more complicated example, the complement of the Cantor set (if you've learned about that) — between any two connected components is another connected component. Interestingly, all of those examples only had at most countably many connected components… – Akiva Weinberger Aug 18 '15 at 19:48
  • @columbus8myhw I am not comfortable with concept of connectedness now. But first example is very good. Thanks. – divya Aug 18 '15 at 19:54
  • @divya In $\mathbb R$, a connected set is just an interval. The Cantor set is just weird. – Akiva Weinberger Aug 18 '15 at 19:55

0 Answers0