How does the topology work of a Klein bottle embedded in $\mathbb{R}^3$ at the points of self intersection?

Question

How does the topology work of a Klein bottle embedded in $\mathbb{R}^3$ at the points of self intersection? Wouldn't the open neighborhoods look like copies of $\mathbb{R}^4$ there instead of $\mathbb{R}^2$? Thanks!

Answer 1

The Klein bottle $K$ does nor embed into $\mathbb R^3$. You only have an immersion $i : K \to \mathbb R^3$ which is locally an embedding. However, $i(K)$ is not a submanifold of $\mathbb R^3$ because it has self intersections. Each intersection point has a neighborhood $U$ such that $U \cap i(K)$ looks like the union of two perpendicular planes in $\mathbb R^3$.