A Visual Introduction to Morse Theory

We can thus conclude:

Morse theory uses the term critical points, or critical values (since technically, $c$ determines the superlevel set but is not a point of the manifold) for the local extrema; all other points are thus considered to be regular points. The following example shows a slightly more complex manifold and illustrates all the sets we encountered so far:

From left to right, we see the level set, the superlevel set, and the sublevel set for the same critical value $c$. Since $\mathbb{M}$ has no interior here, superlevel and sublevel sets can also be seen as filling the manifold with water—Morse theory, in this setting, then describes how different contours merge and split, as the water level is varied .

Source: bastian.rieck.me