let $A$ be a subset of $\mathbb R$ s.t. both $A$ and $\mathbb R-A$ is dense in $\mathbb R$. > Then show that $A$ is nowhere locally compact.

Question

let $A$ be a subset of $\mathbb R$ s.t. both $A$ and $\mathbb R-A$ is dense in $\mathbb R$.

Then show that $A$ is nowhere locally compact.

Answer 1

If $B=A \cap [a,b]$ is compact then it is closed and since $A$ is dense we have $B=[a,b]$ This contradicts that the complement of $A$ is dense.