Smith space

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

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

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

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

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