mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...

, Netto's theorem states that

continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...

bijections of

smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One m ...

s preserve

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...

. That is, there does not exist a continuous bijection between two smooth manifolds of different dimension. It is named after

Eugen Netto Eugen Otto Erwin Netto (30 June 1848 – 13 May 1919) was a German mathematician. He was born in Halle and died in Giessen. Netto's theorem, on the dimension-preserving properties of continuous bijections, is named for Netto. Netto publishe ...

. The case for maps from a higher-dimensional manifold to a one-dimensional manifold was proven by Jacob Lüroth in 1878, using the

intermediate value theorem In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval. This has two im ...

to show that no manifold containing a topological

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...

can be mapped continuously and bijectively to the

real line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...

. Both Netto in 1878, and

Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ; – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of ...

in 1879, gave faulty proofs of the general theorem. The faults were later recognized and corrected. An important special case of this theorem concerns the non-existence of continuous bijections from one-dimensional spaces, such as the

unit interval In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analys ...

, to two-dimensional spaces, such as the

Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions ...

unit square In mathematics, a unit square is a square whose sides have length . Often, ''the'' unit square refers specifically to the square in the Cartesian plane with corners at the four points ), , , and . Cartesian coordinates In a Cartesian coordina ...

. The conditions of the theorem can be relaxed in different ways to obtain interesting classes of functions from one-dimensional spaces to two-dimensional spaces: *

Space-filling curve In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square (or more generally an ''n''-dimensional unit hypercube). Because Giuseppe Peano (1858–1932) was the first to discover one, spa ...

s are surjective continuous functions from one-dimensional spaces to two-dimensional spaces. They cover every point of the plane, or of a unit square, by the image of a line or unit interval. Examples include the

Peano curve In geometry, the Peano curve is the first example of a space-filling curve to be discovered, by Giuseppe Peano in 1890. Peano's curve is a surjective, continuous function from the unit interval onto the unit square, however it is not injective ...

and

Hilbert curve The Hilbert curve (also known as the Hilbert space-filling curve) is a continuous fractal space-filling curve first described by the German mathematician David Hilbert in 1891, as a variant of the space-filling Peano curves discovered by Giusepp ...

. Neither of these examples has any self-crossings, but by Netto's theorem there are many points of the square that are covered multiple times by these curves. *

Osgood curve In mathematical analysis, an Osgood curve is a non-self-intersecting curve that has positive area. Despite its area, it is not possible for such a curve to cover a convex set, distinguishing them from space-filling curves. Osgood curves are named ...

s are continuous bijections from one-dimensional spaces to subsets of the plane that have nonzero

area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while ''surface area'' refers to the area of an open su ...

. They form

Jordan curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition tha ...

s in the plane. However, by Netto's theorem, they cannot cover the entire plane, unit square, or any other

convex set In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...

. *If one relaxes the requirement of continuity, then all smooth manifolds of bounded dimension have equal

cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...

, the

cardinality of the continuum In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by \mathfrak c (lowercase fraktur "c") or , \ma ...

. Therefore, there exist discontinuous bijections between any two of them, as

showed in 1878. Cantor's result came as a surprise to many mathematicians and kicked off the line of research leading to space-filling curves, Osgood curves, and Netto's theorem. A near-bijection from the unit square to the unit interval can be obtained by interleaving the digits of the decimal representations of the

Cartesian coordinate A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured i ...

s of points in the square. The ambiguities of decimal, exemplified by the two decimal representations of 1 = 0.999..., cause this to be an

injection Injection or injected may refer to: Science and technology * Injective function, a mathematical function mapping distinct arguments to distinct values * Injection (medicine), insertion of liquid into the body with a syringe * Injection, in broadca ...

rather than a bijection, but this issue can be repaired by using the

Schröder–Bernstein theorem In set theory, the Schröder–Bernstein theorem states that, if there exist injective functions and between the sets and , then there exists a bijective function . In terms of the cardinality of the two sets, this classically implies that if ...

References

{{reflist, refs= {{citation , last = Dauben , first = Joseph W. , doi = 10.1016/0315-0860(75)90066-X , journal = Historia Mathematica , mr = 476319 , pages = 273–288 , title = The invariance of dimension: problems in the early development of set theory and topology , volume = 2 , year = 1975 {{citation , last = Gouvêa , first = Fernando Q. , doi = 10.4169/amer.math.monthly.118.03.198 , issue = 3 , journal = The American Mathematical Monthly , jstor = 10.4169/amer.math.monthly.118.03.198 , mr = 2800330 , pages = 198–209 , title = Was Cantor surprised? , url = https://www.maa.org/was-cantor-surprised , volume = 118 , year = 2011 {{citation , last = Sagan , first = Hans , doi = 10.1007/978-1-4612-0871-6 , isbn = 0-387-94265-3 , mr = 1299533 , publisher = Springer-Verlag , location = New York , series = Universitext , title = Space-filling curves , year = 1994, url = http://cds.cern.ch/record/1609380. For the statement of the theorem, and historical background, see Theorem 1.3, p. 6. For its proof for the case of bijections between the

and a two-dimensional set, see Section 6.4, "Proof of Netto's Theorem", pp. 97–98. For the application of Netto's theorem to self-intersections of space-filling curves, and for Osgood curves, see Chapter 8, "Jordan Curves of Positive Lebesgue Measure", pp. 131–143. Dimension theory Theorems in topology