Exercise 3.4.1

Question

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

Ulisse Mini · Accepted Answer

Recall a perfect set is a closed set with no isolated points. Thus the intersection of a closed set $P$ and a closed bounded set $K$ gives a closed bounded (and thus compact) set $P \cap K$.

Now take $P = \mathbf{R}$, we get $P \cap K = K$ which is not nessesarily perfect.

Exercise 3.4.1

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Exercise 3.4.1

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