A "torus" is a mathematical abstraction of the surface of a donut. To a topologist, a torus can be built up by "gluing" disks along parts of their boundaries. A "Morse function" gives one way to specify the gluing. Slice the torus by height as in the image. For each cutting plane, we focus our attention on the points of the torus below the plane, namely on "sublevel sets" of the height function. Just above the lowest point of the torus, the sublevel set is a topological disk. Just above the highest point, the sublevel set is the entire torus. As the cutting plane moves upward, the sublevel sets stay topologically unchanged except when the cutting plane is tangent to the torus, i.e., at a "critical value" of the height. Each time the cutting plane passes through a critical value, the sublevel set changes by "attaching" a topological disk along part of its boundary. Morse theory details this gluing in terms of the number of downhill directions of the height function at the critical point where the cutting plane is tangent to the torus.
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