Topological properties of regular and critical points and values

Question

Let $f\colon M\rightarrow N$ be a smooth map between smooth manifolds.

Consider the following two statements, the second one under the assumption

The set of regular points of $f$ are open in $M$, the critical ones are closed in $M$.
The set of regular values of $f$ are open in $N$, the critical ones are closed in $N$.

I think the first statement holds in complete generality, the second one under the assumption that $f$ is a closed map, e.g. a proper one.

Am I correct?

I don't know the book you are using, but a good book to start with is this one http://webmath2.unito.it/paginepersonali/sergio.console/Dispense/Milnor%20Topology%20from%20%23681EA.pdf — Antonio Alfieri, May 19 '15 at 13:47

score 3 · Answer 1 · answered Jul 01 '19 at 18:57

As @AntonioAlfieri remarked in the comments under the question, yes you are correct:

The set of critical points of a smooth map is closed. The set of regular points is defined to be the complement of the set of critical points, so it is open. (You can also take a look at this question.)

Next, suppose that the given smooth map of manifolds is proper (and hence closed).

The set of critical values is defined to the image of the set of critical points. Thus, it is the image of a closed set, and hence it is closed. The set of regular values is defined to be the complement of the set of critical values. So, the set of regular values is open.

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