Exercise 2.28

Question

Exercise 28: Every closed set in a separable metric space is the union of a perfect set and 
    a set which is at most countable.

ghostofgarborg · Accepted Answer

\def\ints{\mathbb{Z}}

We observe that the proof in ex. 27 goes through for any separable metric space. 
    Assume that $E$ is closed. Since $P$ consists of limit points of $E$, $P \cap E = P$. 
    We can therefore write $E = P \cup (P^c \cap E)$, which is a union of a perfect set and 
    a set which is at most countable.

Exercise 2.28

Answers

Comments

Exercise 2.28

Answers

Comments

Add answer