Concept

Homeomorphism

A continuous correspondence between two spaces whose inverse is continuous too. It is the notion of sameness topology uses, and it preserves exactly what is defined from open sets — connectedness, holes, dimension — and nothing defined from distance.

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

A square's worth of points, on a line. A unit square with a point marked, the decimal places of its two coordinates woven into one number, and that number marked on a line beneath.

A line with as many points as a square

Interleave the decimal places of two numbers and one number comes out; take every other place back and the two return. The square has no more points than the segment, and dimension turns out to be invisible to counting.

logic · cardinality

The tent map and the logistic map, joined by a change of coordinate. Two cobweb diagrams side by side — the tent map at slope two and the logistic map at four — with the orbit of one carried to the orbit of the other by a curve drawn between them.

The same map in different coordinates

The tent map and the logistic map at four look nothing alike and are the same map, carried onto each other by a change of variable. Everything either one does the other does, and the change of variable is a sine squared.

dynamics · iteration

Named alongside it

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

Bijection Cardinality Change of variable Conjugacy Continuity Counterexample Decimal expansion Dimension Interleaving Invariance of domain Iteration Logistic map

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