A Small Subset of the Reals

Monet Let {qn} be an enumeration of the rationals, and for each n, let Un be an open interval of length 1/2n centered at qn. Denote the union of all Un 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.

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