For example, if a circle is mapped into the plane as a figure-8, then the crossing is a double point. If a surface is mapped into space so that there is self-intersection, the points of self-intersection are formed by double-points.
Generically, the double-points of a surface in space form curves. For an immersion, they will form closed curves. These curves are called the double curves or the double locus of the mapping, and the double set is the set of points of the surface that map to the double locus.
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The Geometry Center