In mathematics, and especially differential topology,

functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...

and

singularity theory In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it ...

, the Whitney topologies are a countably infinite family of

topologies In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...

defined on the set of smooth mappings between two

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 ma ...

s. They are named after the American mathematician

Hassler Whitney Hassler Whitney (March 23, 1907 – May 10, 1989) was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, characteristic classes, and geometric integratio ...

Construction

Let ''M'' and ''N'' be two real, smooth manifolds. Furthermore, let C^∞(''M'',''N'') denote the space of smooth mappings between ''M'' and ''N''. The notation C^∞ means that the mappings are infinitely differentiable, i.e. partial derivatives of all orders exist and are

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 ...

Whitney C^''k''-topology

For some

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...

, let J^''k''(''M'',''N'') denote the ''k''-jet space of mappings between ''M'' and ''N''. The jet space can be endowed with a smooth structure (i.e. a structure as a C^∞ manifold) which make it into a topological space. This topology is used to define a topology on C^∞(''M'',''N''). For a fixed

consider an open subset and denote by ''S^k''(''U'') the following: :

S^k(U) = \ .

The sets ''S^k''(''U'') form a

basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...

for the Whitney C^''k''-topology on C^∞(''M'',''N'')., p. 42.

Whitney C^∞-topology

For each choice of , the Whitney C^''k''-topology gives a topology for C^∞(''M'',''N''); in other words the Whitney C^''k''-topology tells us which subsets of C^∞(''M'',''N'') are open sets. Let us denote by W^''k'' the set of open subsets of C^∞(''M'',''N'') with respect to the Whitney C^''k''-topology. Then the Whitney C^∞-topology is defined to be the topology whose

is given by ''W'', where: :

W = \bigcup_^ W^k .

Dimensionality

Notice that C^∞(''M'',''N'') has infinite dimension, whereas J^''k''(''M'',''N'') has finite dimension. In fact, J^''k''(''M'',''N'') is a real, finite-dimensional manifold. To see this, let denote the space of

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...

s, with real coefficients, in ''m'' variables of order at most ''k'' and with zero as the constant term. This is a real

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...

with dimension :

\dim\left\ = \sum_^k \frac = \left( \frac - 1 \right) .

Writing then, by the standard theory of vector spaces and so is a real, finite-dimensional manifold. Next, define: :

\implies \dim\left\ = n \dim \left\ = n \left( \frac - 1 \right) .

Using ''b'' to denote the dimension ''B''^''k''_''m'',''n'', we see that , and so is a real, finite-dimensional manifold. In fact, if ''M'' and ''N'' have dimension ''m'' and ''n'' respectively then: :

\dim\!\left\ = m + n + \dim \!\left\ = m + n\left( \frac\right).

Topology

Given the Whitney C^∞-topology, the space C^∞(''M'',''N'') is a

Baire space In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior. According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are e ...

, i.e. every

residual set In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is called ...

dense Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...

., p. 44.

References

{{Reflist Differential topology Singularity theory