What properties must a utility function have such that we can define level sets and thus indifference curves? I asked on Math SE a bit about level sets here. 

Based on what I learned, it seems like we usually assume a level set $L(f)$ and its function $f$ defined on $\Bbb{R}^n$ have the following properties:


$f$ is continuous 
Thus, the curves in the level set are closed curves. (not sure why) 
All closed curves are "converging" to a point. (not sure why) 
This point will be a maximum for the function: the derivative in the point will be zero, since if it was not then the point would be a 1-manifold, which is absurd. 
$\nabla f \neq 0$ on the rest of the points 


I don't understand how we know a preference relation on a consumption bundle has a utility function representation such that these properties hold. According to my professor, we use level sets to describe indifference curves. 

Therefore, if indifference curves are described by a level set, that level set must have the properties listed above. Note, for economics, we are considering a function $U(\mathbf{x})$, where $\mathbf{x}=(x_1,...,x_n)$, and the level set $L(U)$. Also, note $U: \Bbb{R}^n\rightarrow \Bbb{R}$. 

My Question:

In economic terms, what assumption do we have to make to ensure, $U$, $L(U)$ have these properties to create a level set used for indifference curves? 

Ideally, it would be great to have an enumerated list of the assumptions (ie 1,2,3).
Level sets are always well defined. No property of utility functions has to be assumed. For any utility level $\bar u$ just define the level set to be $\{x\in \mathbb{R}^n|U(x)=\bar u\}$. No property of $U$ must be assumed. This is well-defined regardless of the properties of $U$. Properties of $U$ are useful if one wants to derive specific properties of the level sets, but are not needed to define them.