0
$\begingroup$

Can any countable set in $\mathbb{R}$ be extended to a perfect set $P$ in $\mathbb{R}$ such that $R-P$ is uncountable?

$\endgroup$
  • $\begingroup$ I guess you can extend it to $\mathbb{R}$ $\endgroup$ – Yanko Jun 3 '17 at 9:51
2
$\begingroup$

Let $C$ be a countable subset of $\Bbb R$. By definition, a perfect set of a topological space is closed, so if $C$ is dense in $\Bbb R$ then it cannot be extended to a perfect set $P$ with non-empty complement. Conversely, if $C$ is not dense in $\Bbb R$ then there is a non-empty (and, hence, uncountable) open interval $U$ of $\Bbb R$, disjoint from $C$ and we may put $P=\Bbb R - U$.

$\endgroup$
1
$\begingroup$

$Q$ cant be extended as any perfect set containing it will be $R$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.