Is the "vanishing set" the same as the "kernel" of a function? I've just read of a set called "vanishing set" in my topology lecture notes.
It seems to be the kernel of a special type of functions.
Definitions
The lecture notes are in German, but I try to translate them properly:
Vanishing set
Let $F: \mathbb{R}^n \rightarrow \mathbb{R}$ be a function which is continuously differentiable and $X = V(F) := \{x \in \mathbb{R}^n| F(x) = 0\}$ the vanishing set.
Kernel
Let $X,Y$ be groups with neural elements $e_X, e_Y$ and $f: X \rightarrow Y$ a be function. Then $\{x \in X | f(x) = e_Y\}$ is the Kernel of $f$.
Question
First of all, I would like to know if my definitions (especially the of the "vanishing set") is correct. Is it really only for functions $F: \mathbb{R}^n \rightarrow \mathbb{R}$?
Secondly, I would like to know the difference of "Kernel" and "Vanishing set".
If the vanishing set is only defined for functions $F: \mathbb{R}^n \rightarrow \mathbb{R}$, then the answer is clear. But as my first definition was only a note of a remark the professor made, I'm not too sure about it.
 A: I assume you are asking about standard mathematical English usage of words. Your definitions are fine, and like all definitions can be made more general by those seeking greater generality.
Vanish is used to mean something is 0. A function $f$ vanishes at $x$ means that $f(x)=0$, a function $f$ vanishes on the set $X$ means that $f(x)=0$ for all $x \in X$. This use of the symbol $0$ is typically required: if it merely equals some "neutral" element, then the word "vanish" is not used.
Kernel is used in algebra and category theoretic areas to mean something that describes those pairs $(x,y)$ so that $f(x)=f(y)$ where $f$ is a nice function, a homomorphism, or a morphism. It is not used when $f$ is merely continuously differentiable (rather when it is a linear function, for instance $f(x_1,x_2,\ldots,x_n) = \sum a_i x_i$ for some fixed real numbers $a_i$). In some parts of algebra (group theory, ring theory, module theory, for example) the kernel can be described very compactly by saying that $f(x) = f(y)$ iff $f(x-y) = 0$ and so the kernel is just $\{ x : f(x) = 0 \}$. Here the symbol $0$ is not as important, especially in group theory, where it may be denoted as $1$ or $e$.
Kernel is used in analysis to mean something completely unrelated: it is a function to be convoluted against. For instance, see the pretty pictures in this wikipedia article.
In particular, if someone used the word kernel to describe the vanishing set of $\sin(x)$ it would cause confusion in most audiences.
The difference between a kernel and a vanishing set actually comes up in character theory of finite groups where the kernel of a homomorphism from $G$ to a matrix group are those group elements that are sent to the identity matrix, but the vanishing set is all those group elements that are sent to any matrix of trace zero (the “character” is the trace, hence the vanishing set refers to the character being zero).
I checked the wikipedia article on the zillions of different meanings of kernel. In mathematics, I think they all come down to one of those two. The German word “Kern” has some other meanings too in mathematics (often translated as “core”).
