Compactness
Named by 10 essays across 5 fields — each of them below, with the objects they name alongside it.
An infinite tree has an infinite path
A tree that goes on forever, in which every node has only finitely many children, must contain a single branch that goes on forever. The proof is a rule for walking, and the rule is the whole of why finite information can decide an infinite question.
Three in a row on the number line
Colour the numbers one to eight in two colours and it can be arranged that no three equally spaced numbers agree. Add the ninth and it cannot. The structure being forced is arithmetic rather than graphical, and the proof is a different proof.
One chart is never enough
Stereographic projection matches the sphere minus a point with the whole plane, and the missing point is not a blemish to be tidied away. It is a theorem — no single flat picture covers a sphere — and the repair is two pictures with a rule for passing between them.
The subsequence that has to exist
Every bounded list of numbers has a part that settles down. A bounded list of functions need not: the waves sin 2πkx never come within 1.76 of one another. One extra condition — that no member may change faster than a bound they all share — restores the guarantee, and it is the reason a differential equation with a continuous rule has a solution at all.
A number larger than every number
Ask for a number bigger than 0, bigger than 1, bigger than 2, and so on for ever. Every finite piece of that request is granted by an ordinary number, so compactness grants all of it at once — in a structure that satisfies every sentence true of the whole numbers and still contains something beyond all of them. Nothing in first-order logic can say 'and nothing else'.
Every chord slid to the middle
Take a shape, pick a line, and slide every chord that crosses the line at right angles until the line cuts it in half. The area cannot change, the boundary can only get shorter, and the result is symmetric. Do it again about another line, and another, and the shape is squeezed towards a disc — unless the lines are badly chosen, in which case it stops short.
Enough partners in every finite group
Hall's theorem says a matching exists unless some group has too few partners between them. On an infinite graph that is false: one vertex that can be paired with anybody, and every other with exactly one, passes the test on every finite group and has no matching at all. The failure needs a vertex with infinitely many choices. Forbid that, and the theorem comes back, by an argument about trees rather than about matchings.
How many colours the plane needs
Colour every point of the plane so that no two points exactly one unit apart share a colour. Seven colours are enough, by a tiling with hexagons, and four are necessary, by a graph of seven points. For sixty-eight years nothing better was known on either side; then in 2018 a graph of 1,581 points showed four is not enough — and the answer is still somewhere from five to seven.
Closed sets obey the continuum hypothesis
Whether some set of reals is bigger than the whole numbers and smaller than the line is a question the axioms cannot answer. For closed sets it has an answer, found in 1883: strip away the isolated points, again and again, and what is left is either nothing or a perfect set, which is as large as the line — so every closed set is countable or of the line's size. The same answer holds for Borel and analytic sets, and fails to be provable exactly one step further up.
![Polynomials climbing to the square root, each above the last. The iterates of p ↦ p + (x − p²)/2 at steps 1, 2, 3, 4, 6, 9 on [0, 1] beneath the curve √x, with largest gaps 1.000, 0.500, 0.259, 0.176, 0.108, 0.068.](/f/polynomials-climbing-to-the-square-root-each-above-the-last-ce601e49.svg)
Settling in order settles everywhere at once
A sequence of functions can settle at every point and never settle uniformly — the largest gap stays put while each point escapes it. Dini found the circumstances in which that cannot happen: continuous functions, a continuous limit, every member above the next, and a closed interval. Under those four conditions convergence at each point is convergence everywhere at once, and Weierstrass's iteration for the square root becomes a sequence of polynomials converging uniformly to |x|.
Named alongside it
The objects these essays reach for when they reach for this one.
Existence proofAxiom of choiceConsistencyContinuityExhaustive searchGraph colouringPigeonhole principleSupremumTilingUniform convergenceAreaArithmetic progression