I would like to have some explanation about a fact which is written in the first chapter of Milnor's book "topology, from the differentiable viewpoint". The fact is the following: Suppose $M,N$ be two differentiable manifolds of the same dimension. Let $f: M \rightarrow N$ be a differentiable map with $M$ compact and let $y \in N$ be a regular value. We define #$f^{-1}(y)$ to be the number of points in $f^{-1}(y)$, thus $f^{-1}(y)$ does not contain any critical point. And now the part which I do not understand: the book says that an observation that can be made about #$f^{-1}(y)$ is that is locally constant as a function of $y$ (where $y$ ranges only through regular values). I don't understand how to figure out this fact, I tough considering #$f^{-1}: f^{-1}(y) \rightarrow \mathbb{N}$, but still remains unclear to me why it should be a locally constant function. Could someone please can explain to me this fact maybe also (if I am not asking too much) giving a simple example.
This is a doubt from Milnor's "Topology from a Differentiable Viewpoint".     For a smooth $f:M\to N$, with $M$ compact, and a regular value $y\in N$, we define $n(f^{-1}(y))$ to be the number of points in $f^{-1}(y)$. It is easy to see that $n(f^{-1}(y))$ is finite. It can also be seen that $n(f^{-1}(y))$ is locally constant; i.e. there is a neighbourhood of $y$ such that for every point $y'$ in that neighbourhood, $n(f^{-1}(y'))=n(f^{-1}(y))$.   The proof for the statement in bold is the following:     Let $x_1,x_2,\dots,x_n$ be the points of $f^{-1}(y)$, and choose disjoint neighbourhoods $U_1,U_2,\dots,U_n$ of these points, which are mapped diffeomorphically into neighbourhoods $V_1,V_2,\dots,V_n$ of $y$. The required neighbourhood is $(V_1\cap V_2\cap\dots V_n)-f(M-U_1-U_2-\dots U_n)$.   I don't understand how $(V_1\cap V_2\cap\dots V_n)-f(M-U_1-U_2-\dots U_n)$ is open. I know that $(V_1\cap V_2\cap\dots V_n)$ is open. However, how can one prove that $f(M-U_1-U_2-\dots U_n)$ is closed? I can see this working if $f$ is surjective, but not otherwise.
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