Number of connected components in a set

Question

I have a function $H$ with canonical coordinates $(q,p)$ on the cotangent bundle $T^*Q \cong S^1 \times \mathbb{R}$. The function $H$ has 4 critical points and has regular values where ever it is not zero. Call the set of regular values $U \subset \mathbb{R}$.

How do I tell how many connected components this set has?

Answer 1

The set of regular values is connected, i.e. it has one component.

All you need to prove this are two facts and one theorem:

1. $H$ is a continuous function.
2. The set $C = (S^1 \times \mathbb R) - (\text{critical points})$ is a connected set

This is true because $S^1 \times \mathbb R$ is a connected 2-manifold, and the complement of a finite set in any $n$-manifold (with $n \ge 2$) is connected.

The theorem you need is:

3. The continuous image of a connected set is connected.

That is a theorem of topology, and it doesn't care how many dimensions the domain and range have, all that matters is that the domain is a connected topological space.

Since the set of regular values is the $H$ image of $C$, they form a connected set.