0
$\begingroup$

I have been reading the book "Introduction to Set Theory" by Jech and Hrbacek and have come to the following exercise in the chapter on sets of real numbers :

Every perfect set is either an interval of form $[a,b]$, $(-\infty,a]$, $[b,\infty)$, or $\boldsymbol{R} = (-\infty,\infty)$, or it is the union of two disjoint perfect sets.

I am able to show that the first four intervals are perfect and cannot be written as a disjoint union of perfect sets. I haven't been able to yet show that if a perfect set is not an interval of one of the first four types that it can be written as a union of two disjoint perfect sets. It is easy to see that a union of closed intervals is a perfect set that meets the requirement that it is a union of two disjoint perfect sets. A union of closed intervals with $(-\infty,a]$ and $[b,\infty)$ would also meet this requirement.

Is it true that any perfect set that is not one of the four interval types above is a disjoint union of intervals of the first three types of intervals listed in the exercise above ? If this is the case how can it be shown ? Also, if it is not the case, can anyone help with the last part of the exercise.

$\endgroup$
3
  • $\begingroup$ The Cantor set. $\endgroup$
    – Asaf Karagila ♦
    Commented Nov 9, 2021 at 19:14
  • $\begingroup$ Mind you, the exercise only requires you to show that, if your perfect set $P$ is not one of those four types of sets, then it can be represented as $P=Q\cup R$, with $Q, R$ perfect, $Q\cap R=\emptyset$. It does not require you to show that $Q$ and $R$ themselves must be intervals of any sort. Hint: prove that there is a point $x\in\mathbb R$ such that $x\not\in P$ and that $Q=P\cap (-\infty, x), R=P\cap (x, +\infty)$ are both perfect. $\endgroup$
    – user700480
    Commented Nov 9, 2021 at 19:18
  • $\begingroup$ Every set can be written as the union of closed intervals trivially - $A = \bigcup\limits_{a \in A} [a, a]$. $\endgroup$
    – Mark Saving
    Commented Nov 9, 2021 at 21:37

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .