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. Whitney Ck-topology For some integer , let Jk(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 consider an open subset and denote by Sk(U) the following: The sets Sk(U) form a basis for the Whitney Ck-topology on C∞(M,N). Whitney C∞-topology For each choice of , the Whitney Ck-topology gives a topology for C∞(M,N); in other words the Whitney Ck-topology tells us which subsets of C∞(M,N) are open sets. Let us denote by Wk the set of