Let {q_{n}} be an enumeration of the rationals, and for each n, let U_{n} be an open interval of length 1/2^{n} centered at q_{n}. Denote the union of all U_{n} by U. Then U is a dense open subset of the reals which has measure at most 1.

From a topological point of view, this certainly seems to be a reasonable observation. After all, the rationals are dense and have measure zero, so it isn’t surprising that by making the measure positive one can add the condition of openness. But from a more direct perspective, the construction seems to yield a very strange set–a spattering of droplets across the real line which cover very little length but come arbitrarily close to any point. Like an Impressionist painting, one must add a little distance to make out the comprehensible form.

Image by Claude Monet.

COMMENT!!!

Happy Thanksgiving Bryan! Your blog is very nice.

A very sexy art+math fusion!