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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 3 essays across one field — 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

Named alongside it

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

MeasureMeasure zeroCountabilityLimitAxiom of choiceCantor setContinuityContradictionEquivalence relationGeometric seriesInfimumInterval

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