In mathematics, and especially differential topology

Differential topology

In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...

, functional analysis

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 and the linear operators acting upon these spaces and respecting these structures in a suitable sense...

 and singularity theory, the Whitney topologies are a countably infinite family of topologies defined on the set of smooth mappings between two smooth manifolds. They are named after the American mathematician Hassler Whitney

Hassler Whitney

Hassler Whitney was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, and characteristic classes.-Work:...

.

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 derivative

Partial derivative

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant...

s of all orders exist and are continuous

Continuous function

In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

.

Whitney C^k-topology

For some integer

Integer

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

 , let J^k(M,N) denote the k-jet space

Jet (mathematics)

In mathematics, the jet is an operation which takes a differentiable function f and produces a polynomial, the truncated Taylor polynomial of f, at each point of its domain...

 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 integer

Integer

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

  consider an open subset and denote by S^k(U) the following:
The sets S^k(U) form a basis for the Whitney C^k-topology on C^∞(M,N).

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 basis is given by W, where:

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

Polynomial

In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

s, with real coefficients, in m variables and of order at most k. This is a real vector space

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

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

Topology

Consider the surjective mapping from the space of smooth maps between smooth manifolds and the k-jet space:
In the Whitney C^k-topology the open sets in C^∞(M,N) are, by definition, the preimages of open sets in J^k(M,N). It follows that the map π^k between C^∞(M,N) given the Whitney C^k-topology and J^k(M,N) given the Euclidean topology is continuous.

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

Baire space

In mathematics, a Baire space is a topological space which, intuitively speaking, is very large and has "enough" points for certain limit processes. It is named in honor of René-Louis Baire who introduced the concept.- Motivation :...

, i.e. every residual set is dense.