The Mandelbrot set is bizarre, astoundingly so. E.g., since it is a fractal, it is astoundingly irregular. Mandelbrot argued that commonly shapes in nature ar

The Mandelbrot set is bizarre, astoundingly so. E.g., since it is a fractal, it is astoundingly irregular. Mandelbrot argued that commonly shapes in nature are fractals. So, we could go to a mountain range, find a lake, and then suspect that the surface is a 2D fractal. Well, as bizarre as the Mandelbrot set is, actually it is a closed set in the usual topology for the plane. Then there is a theorem that there is a function from the plane to the line that is 0 on the Mandelbrot set, strictly positive otherwise, and infinitely differentiable. Similarly there is a function that is 0 on the boundary of that set, negative in the interior, and positive otherwise. So, in general, with the Mandelbrot set just an example, these results hold for any closed set in the plane. And the result generalizes to dimension 1 and all finite dimensions greater than 2. So, a set is closed if and only it it is the level set of an infinitely differentiable function 0 on the boundary of the set, negative in the interior, and positive otherwise. This result can be used to show that in the constraint qualifications for the Karush-Kuhn-Tucker conditions, the (i) Zangwill and (ii) Kuhn-Tucker constraint qualifications are independent. For a while that fact was an outstanding unanswered question about constraint qualifications. Somewhere in one of the papers of Arrow, Hurwicz, and Uzawa on mathematical economics there is a similar question about constraint qualifications asked but not answered that is easily answered with an example based on these results about infinitely differentiable functions and closed sets. Back to the mountains, we can argue that the surface of the land, above, at the boundary, and on the bottom of the lake can be infinitely differentiable while the lake is the Mandelbrot set or any closed set. Bizarre stuff!