0
$\begingroup$

Let $M$ be a smooth manifold. Let's call $M$ quasi-seperated if $M$ has the following property: If $B,C \subseteq M$ are open balls, then $B \cap C \subseteq M$ is a finite(!) union of open balls. By an open ball I mean an open submanifold, which is diffeomorphic to some $D^n$.

Is every manifold quasi-separated? If not, are open balls quasi-separated? Is there a simple characterization of quasi-separated manifolds?

Improve this question
$\endgroup$
4
  • 1
    $\begingroup$ In the plane you can find two open subsets whose intersection has infinitely components so, in particular, they do not intersect in finitely many balls. $\endgroup$
    – Mariano Suárez-Álvarez
    Sep 2, 2010 at 15:58
  • $\begingroup$ Could you give an example of a quasi-separated manifold? Where does this come from? It seems to me that a smooth manifold is quasi-separated if and only if $R^n$ is. $\endgroup$
    – Deane Yang
    Sep 2, 2010 at 16:00
  • 1
    $\begingroup$ (Foe example, let one of the sets be the open upper half plane unioned with all open balls of radius $1/4$ centered at the points $(n,0)$ with $n\in\ZZ$, and the other set be the reflection of the first one on the $x$-axis.) $\endgroup$
    – Mariano Suárez-Álvarez
    Sep 2, 2010 at 16:00
  • $\begingroup$ @Mariano: I already had this example in mind, but I had some doubt that this set is actually an open ball. Thus we can conclude that every quasi-separated manifold is $1$-dimensional, and vice versa? $\endgroup$
    – Martin Brandenburg
    Sep 2, 2010 at 16:54

1 Answer 1

Reset to default
1
$\begingroup$

Using the Riemann mapping theorem you can easily show that $\mathbb{R}^2$ is not quasi separated.

Improve this answer
$\endgroup$
1
  • $\begingroup$ Oh, I see that Mariano already posted that as a comment. $\endgroup$
    – Michael Bächtold
    Sep 2, 2010 at 16:00

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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