Concept

Invariance of domain

The theorem that a continuous one-to-one map from an open piece of n-dimensional space into n-dimensional space has an open image. Its consequence is that spaces of different dimensions are never homeomorphic, which is what makes dimension a property of a space rather than of a way of describing it.

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

A ray through a knotted tube, crossing it 5 times. A closed surface in space — a tube round a trefoil knot — with a point, a ray from it and every crossing of the surface marked; the parity of the count says which side of the surface the point is on.

Two pieces, in every dimension

A closed curve cuts the plane in two. A closed curve in space cuts nothing at all, and it takes a closed surface to do the job — which is the shape of the general theorem, and the reason the word "dimension" means anything.

topology · jordan curve

Named alongside it

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

Dimension Homeomorphism Bijection Cardinality Closed curve Connectedness Continuity Counterexample Decimal expansion Degree Interleaving Jordan curve

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