Let $(M, g)$ be a compact connected Riemann manifold without boundary. Let $A \subset M$, be residual, i.e. the intersection of countably many sets with dense interiors. Can there be finitely many open subsets $U_1, ... U_n$ of $A \cap M$, that are also closed, so that

$U_i \cap U_j = \varnothing $ for $i \neq j$ and

$\bigcup_{i=1}^n U_i = A \cap M$


Let $A$ be the complement of the equator on the sphere. It has two open components and is clearly residual.

If you want a set whose components aren't open, remove two sequences from $A$, one that converges to (but never hits) the north pole and one that converges to (but never hits) the south pole.


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.