Regular sublevels are compact manifolds with boundary

If $x\in S$ has $f(x)<c$, continuity of $f$ and openness of $(-\mathrm{\infty },c)$ give an open neighbourhood $U$ of $x$ with $f(U)\subseteq (-\mathrm{\infty },c)$, so $U\subseteq S$. Hence every point of the open sublevel is interior; the local normal form below excludes points of the level from the interior.

Let $x\in f^{-1}(c)$. By [F1] the differential $d{f}_x$ is nonzero, so $f$ is a submersion at $x$; by [L1] choose adapted source coordinates with last coordinate $f-c$. To see that these are coordinates, start with the submersion normal form and replace its final coordinate by ${\psi }^{-1}(u_m)-c$, whose derivative is nonzero. Then $S\cap U$ corresponds to $\{u_m\le 0\}$ and the level to $\{u_m=0\}$. Every neighbourhood of a level point also meets $\{f>c\}$, so no level point is interior to $S$.

The charts of step 1.2 around points of the level together with the ordinary charts of $M$ around the interior points of step 1.1 cover $S$. Their transition maps are restrictions of smooth transition maps of the ambient manifold $M$ (each boundary chart extends to an ambient smooth chart), hence smooth in the local-extension sense; therefore they define a smooth $m$-manifold structure with boundary on $S$ whose boundary is the level $f^{-1}(c)$ and whose interior is the open sublevel, and [L2] makes that boundary closed and embedded.

By hypothesis $S$ is compact. If the level is empty, step 1.1 applies at every point and $S=S^{\circ }$, so $S$ is a compact manifold without boundary; conversely, if the boundary is empty then the level is empty. The assertion for a Morse function with compact sublevels is the special case in which the regular values are exactly the non-critical values.