nLab Smith space (generalized smooth space)

For the concept of the same name in functional analysis see at Smith space (functional analysis).

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 $X$ 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 $f \in \mathcal{F}$.

References

• J. Wolfgang Smith?, The de Rham theorem for general spaces, Tohoku Math. J. (2), Volume 18, Number 2 (1966), 115-137, (project euclid)