Concept

Nowhere dense

A set is nowhere dense when its closure contains no interval — however small a window, the set misses some sub-window entirely. A countable union of such sets is meagre, and Baire's theorem says the whole line is never meagre, which is what makes nowhere-dense sets a usable notion of small.

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

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

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.

Cantor set Baire category Continuity Counterexample Geometric series Interval Limit Measure Measure zero Oscillation Pointwise convergence Rational number

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