Define what is meant by the regular values and critical values of a smooth map $f: X \rightarrow Y$ of manifolds. State the Preimage Theorem and Sard's Theorem.

Suppose now that $\operatorname{dim} X=\operatorname{dim} Y$. If $X$ is compact, prove that the set of regular values is open in $Y$, but show that this may not be the case if $X$ is non-compact.

Construct an example with $\operatorname{dim} X=\operatorname{dim} Y$ and $X$ compact for which the set of critical values is not a submanifold of $Y$.

[Hint: You may find it helpful to consider the case $X=S^{1}$ and $Y=\mathbf{R}$. Properties of bump functions and the function $e^{-1 / x^{2}}$ may be assumed in this question.]