nLab
differential module

Roman Sikorski defined the term “differential module” in 1971/72. Mark Mostow further developed the theory in 1979. The following definition of a differentiable space (a kind of generalized smooth space) comes from Mostow’s paper where it is attributed to Sikorski.

Definition

A differentiable space is a topological space XX together with, for each open UU in XX, a collection C (U)C^\infty(U) of continuous real-valued functions on UU, satisfying the closure conditions:

  1. The rule UC (U)U \to C^\infty(U) defines a sheaf on XX (denoted C XC^\infty X).
  2. For any nn, if f 1,,f nC (U)f_1, \dots, f_n \in C^\infty(U) and gC ( n)g \in C^\infty(\mathbb{R}^n) (with the usual meaning), then g(f 1,,f n)C (U)g(f_1, \dots, f_n) \in C^\infty(U).

The elements of C (X)C^\infty(X) are called smooth functions on XX.

References

Revised on April 16, 2009 23:39:01 by Toby Bartels (71.104.234.95)