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Concept

Dense set

A set meeting every interval of the line, however short, so that no gap in it can be found. Being dense says nothing about how much room a set takes up: the rationals are dense and have measure nought.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

The rationals covered by intervals of total length 0.1800. Intervals of rapidly shrinking length placed around the rationals of the unit interval in the order they are listed, with the union of them drawn as a single band beneath.

Covering a set from outside

To say how long a set is, cover it with intervals and add their lengths, then take the smallest total any covering achieves. That definition is short, obviously right for an interval, and gives the rationals a length of nothing.

analysis · Measure Two indicators, and the upper sum that will not come down. A partition of the unit interval drawn against the middle-thirds set and against a set of positive length, above a chart of each one's upper sum as the partition is refined.

Which functions can be added up

Riemann's integral works when the upper and lower sums close on each other. The exact condition for that, found once measure existed to state it in, is that the points where the function jumps have measure zero — which some nowhere dense sets fail.

analysis · Measure The classes, a selection from them, and the translates that cannot have a length. Points of several classes of the unit interval under translation by rationals, drawn one class per row, above rows showing rational translates of a selection that never overlap.

A set that has no size at all

Slide the unit interval along itself by every rational and the points fall into classes. Choose one point from each and the resulting set has no length — not zero, not positive, none: countably many disjoint copies of it would have total length nought or infinity, and the union needs something in between.

analysis · Measure An additive function that is nowhere a line. A square window with 1157 points of the graph of an additive function that sends √2 to 0, scattered across the window and meeting every part of it, beside a dashed diagonal marking the straight line y = x.

A function that adds and is nowhere a line

Every continuous function with f(x + y) = f(x) + f(y) is a straight line through the origin. Drop continuity and, given the axiom of choice, there are others — functions that add perfectly and whose graphs are scattered densely over the whole plane. A finite piece of the construction can be drawn exactly; the whole of it needs a basis of the real numbers that no one can write down.

logic · Axiom of choice Riemann's function on one period, with its one kind of smooth point. R(x) = Σ sin(n²x)/n² on [0, 2π]; R(π) = 0 with derivative −1/2; R(π/2) = 1.2336, a cusp.

The slope a Gauss sum leaves behind

Riemann is said to have offered sin x + sin(4x)/4 + sin(9x)/9 + … as a continuous function with no derivative anywhere. It has one after all, at π and at every π times an odd number over an odd number, and the slope there is always exactly −1/2. Computed with a million terms, the function's behaviour at every fraction is read off a single number — a quadratic Gauss sum — and the slope appears exactly where that number is zero.

analysis · Uniform convergence

Named alongside it

The objects these essays reach for when they reach for this one.

Measure zeroMeasureAxiom of choiceContinuityCountabilityLimitBasisCantor setContradictionDerivativeDifferentiabilityEquivalence relation

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