The question is :-
Let $K$ be a compact subset of $\mathbb R$ with non empty interior. Show that K is of the form $[a,b]$ or $[a,b] \setminus \bigcup I_n$ , where { $I_n$} is a countale disjoint family of open intervals with end points in K.
Attempt at proof :-
By Heine-Borel theorem , we know that a compact set $K$ in $\mathbb R$ is closed and bounded. Since it is bounded, closed sets of the form $[a,\infty]$ and $[-\infty,a]$ are not compact. Also since $K$ has a non empty interior it can not be a finite set.
Since K is a closed set, it can not be of the form of an arbitrary union of closed sets. Since an arbitrary union of closed sets may not necessarily be closed. Thus , it can be a finite union of closed sets.
A finite union of closed sets can be represented in the form $[a,b] \setminus \bigcup I_n$ .
Also K may have some isolated points, for example it is of the form $[c,d] \bigcup${$p$} , but such sets can also be represented using above form. For example if $p > d$ we can express it as $[c,p] \setminus(d,p)$.
Attempt at proof ends here.
This is sort of the general idea which I am having , but I am certain that I have missed something and I am unable to express it mathematically. Please bare with this I am new to writing proofs. I think in my attempt I have not considered countable {${I_n}$} but rather finite, which is one more error in the proof.Thank You.
Best Answer
Hints.
Prove that any open subset of $\mathbb{R}$ is an (at most) countable union of disjoint open intervals.
Assume temporarily that the result is true. How would you characterize the numbers $a$ and $b$ in terms of the set $K$? (I am asking this question so that when you return to the actual problem, you can say at the outset "Let $a$ be such or such number," and similarly for $b$.)