# Whitney topologies

In mathematics, and especially differential topology, functional analysis 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.

## 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.^{[1]}

### Whitney C^{k}-topology

For some integer *k* ≥ 0, 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 integer *k* ≥ 0 consider an open subset *U* ⊂ J^{k}(*M*,*N*), and denote by *S ^{k}*(

*U*) the following:

The sets *S ^{k}*(

*U*) form a basis for the

**Whitney C**on C

^{k}-topology^{∞}(

*M*,

*N*).

^{[2]}

### Whitney C^{∞}-topology

For each choice of *k* ≥ 0, 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:

^{[2]}

## 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 ℝ^{k}[*x*_{1},…,*x*_{m}] denote the space of polynomials, with real coefficients, in *m* variables of order at most *k* and with zero as the constant term. This is a real vector space with dimension

Writing *a* = dim{ℝ^{k}[*x*_{1},…,*x*_{m}]} then, by the standard theory of vector spaces ℝ^{k}[*x*_{1},…,*x*_{m}] ≅ ℝ^{a}, and so is a real, finite-dimensional manifold. Next, define:

Using *b* to denote the dimension *B*^{k}_{m,n}, we see that *B*^{k}_{m,n} ≅ ℝ^{b}, and so is a real, finite-dimensional manifold.

In fact, if *M* and *N* have dimension *m* and *n* respectively then:^{[3]}

## 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, i.e. every residual set is dense.^{[4]}

## References

- ↑ Golubitsky, M.; Guillemin, V. (1974),
*Stable Mappings and Their Singularities*, Springer, p. 1, ISBN 0-387-90072-1 - 1 2 Golubitsky & Guillemin (1974), p. 42.
- ↑ Golubitsky & Guillemin (1974), p. 40.
- ↑ Golubitsky & Guillemin (1974), p. 44.