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](http://www.mathcounterexamples.net/positive-polynomial-not-reaching-its-minimum/).
* 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](http://www.mathcounterexamples.net/compact-convex-set-whose-extreme-points-set-is-not-closed/).


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



Results true in a dimension and false for higher dimensions
An n-dimensional brownian motion visits every neighborhood of $\mathbb{R}^n$ infinitely often with probability 1 iff $n \leq 2$


