Concept

Baire category

Baire's theorem: a complete space is not a countable union of nowhere-dense sets, so a countable list of small sets cannot exhaust it. It turns questions about typical behaviour into questions about which sets are meagre, and produces objects — continuous functions nowhere differentiable, for instance — without constructing one.

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

The algebraic numbers, arriving in finite batches. A stretch of the number line with the roots of integer polynomials marked, each at the height of the smallest polynomial that catches it, and the count of polynomials at each height.

Countable, and everywhere

The numbers a polynomial can catch arrive in finite batches, so they can be listed. They are also in every interval, however short. Being listable turns out to say nothing whatever about being sparse.

logic · Cardinality $Thomae's function as a limit of tents, at n = 3 and 8. Continuous functions built from narrow triangles over the fractions, drawn at two stages, with the limit shown as a dot at height 1/q over every fraction p/q: a function continuous at the irrationals and discontinuous at the rationals.$

A limit can jump at every fraction

A sequence of continuous functions can settle, point by point, on a function that is discontinuous at every rational number. It cannot settle on one that is discontinuous everywhere — the indicator of the rationals needs two limits in a row, and Riemann's integral cannot follow the second. The line between the two is Baire's theorem, and it measures smallness by gaps rather than by length.

analysis · Uniform convergence

Named alongside it

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

Algebraic number Cantor set Continuity Countability Counterexample Density Enumeration Height Listing Measure zero Nowhere dense Ordering

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