Some theorems are true in vector spaces or in manifolds for a given dimension $n$ but become false in higher dimensions.

Here are two examples:

- A positive polynomial not reaching its infimum. Impossible in dimension $1$ and possible in dimension $2$ or more. See more details here.
- A compact convex set whose set of extreme points is not closed. Impossible in dimension $2$ and possible in dimension $3$ or more. See more details here.

What are other "interesting" results falling in the same category?

infimum, or, nothavinga minimum. $\endgroup$ – Gerry Myerson Mar 6 '17 at 23:18