nLab
Smith space

In 1966, J. Wolfgang Smith studied the following extension of the notion of a smooth manifold.

Definition

A differentiable structure on an arbitrary topological space XX is a family, \mathcal{F}, of real-valued functions on XX satisfying a certain closure condition.

To express the closure condition, we need an auxiliary notion. A plot of (X,)(X, \mathcal{F}) is a continuous map ϕ:UX\phi : U \to X with domain an open subset of some Euclidean space with the property that fϕC (U)f \circ \phi \in C^\infty(U) for all ff \in \mathcal{F}.

The closure condition is that if a continuous map f:Xf: X \to \mathbb{R} has the property that whenever ϕ:UX\phi: U \to X is a plot for (X,)(X, \mathcal{F}) then fϕC (U)f \circ \phi \in C^\infty(U) then ff \in \mathcal{F}.

References

Revised on April 16, 2009 23:39:36 by Toby Bartels (71.104.234.95)