34 Perfect and Connected Sets Practice

#todo/school : Read §3.4 📅 2024-11-05 ✅ 2024-11-06
#todo/school : Do 3.4.1, 2 ✅ 2024-11-06

3.4.1

Question

If $P$ is a perfect set and $K$ is compact, is the intersection $P \cap K$ always compact? Always perfect?

The idea is that $P$ is closed and has no isolated points, while $K$ is closed and bounded. So then an example of a set of $K$ that is compact while having isolated points would be $K = [0, 1] \cup {2}$ . On the other hand, a set $P$ that is perfect but isn't bounded would be $P = [1, \infty)$ . If we do $K \cap P$ :

K \cap P = {1, 2}

Notice that

the set had to be bounded since the set is a subset of a bounded set.
the set had to be closed since the arbitrary intersection of closed sets is closed.
the set got more isolated points, so then it's not perfect now (this is a good counter-example).

Proof

The set must always be compact, because $P \cap K$ being the arbitrary intersection of closed sets is closed. Further, since it is a subset of $K$ which is a bounded set, then it must be bounded too.

Now the set may not be perfect. See using $K = [0, 1] \cup {2}$ and $P = [1, \infty)$ . Then:

K \cap P = {1, 2}

which has isolated points $1, 2$ . Thus it isn't perfect.

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3.4.2

Question

Does there exist a perfect set consisting of only rational numbers?

Proof

No this would be impossible! This is because any such set $P \subseteq Q$ are countable (since $P \subseteq Q \sim N$ ) but Perfect Sets are Uncountable.

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