Let $f:M\rightarrow\mathbb{R}$ be a smooth function, where $M$ is a closed
manifold. I have the feeling that the [Lusternik-Schnirelmann category](https://en.wikipedia.org/wiki/Lusternik%E2%80%93Schnirelmann_category) of $M$
is an estimate from below of the number of *components* of the critical
points set of $f$, rather than only the number of critical points of $f$,
which is, of course, a stronger result. Am I wrong? Any references are welcome.



Is L-S category estimating the number of components of the critical set?
Doesn't the height function on a torus laying down on its side give a counter-exsmple? There are two circles of critical points, but $\mathrm{cat}(T^2) =3$.


*Added:* A constant function on a connected non-contractible manifold gives an even easier example.


