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∞ 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.Whitney C''k''-topology
For some integer, let Jk denote the k-jet space of mappings between M and N. The jet space can be endowed with a smooth structure which make it into a topological space. This topology is used to define a topology on C∞.For a fixed integer consider an open subset and denote by Sk the following:
The sets Sk form a basis for the Whitney Ck-topology on C∞.
Whitney C∞-topology
For each choice of, the Whitney Ck-topology gives a topology for C∞; in other words the Whitney Ck-topology tells us which subsets of C∞ are open sets. Let us denote by Wk the set of open subsets of C∞ with respect to the Whitney Ck-topology. Then the Whitney C∞-topology is defined to be the topology whose basis is given by W, where:Dimensionality
Notice that C∞ has infinite dimension, whereas Jk has finite dimension. In fact, Jk is a real, finite-dimensional manifold. To see this, let 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 dimensionWriting then, by the standard theory of vector spaces and so is a real, finite-dimensional manifold. Next, define:
Using b to denote the dimension Bkm,''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: