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Concept

Interval

A stretch of the line between two endpoints, with or without them. It is the object measure is defined against — every other set's length is an infimum over coverings by intervals — and a set containing none of them is nowhere dense.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

A closed interval and an open one, matched point for point. Two number lines, one closed and one open, with arrows showing the countable sequence of points that has to move.

Two injections make a bijection

If each of two collections fits inside the other without collisions, they are the same size. That sounds obvious and is not, because neither injection needs to be onto — and the proof is a rule for deciding which of the two to follow, one chain at a time.

logic · Cardinality A set with no interval in it and half its length left, after 6 stages. Stages of removing a shrinking middle from every surviving interval, with the total length left printed at each stage, and the middle-thirds construction of the same depth drawn beneath for comparison.

No interval in it, and length to spare

The middle-thirds set has no length because the removed pieces add to one. Remove shrinking middles instead and they add to a half — leaving a set that still contains no interval anywhere, and still has half the length it started with.

analysis · Measure 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

Named alongside it

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

CountabilityGeometric seriesLimitMeasureMeasure zeroBijectionCantor setCardinalityChain decompositionConstructionDense setHilbert hotel

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