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 $X$ is a family, $ℱ$, of real-valued functions on $X$ satisfying a certain closure condition.

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

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

References

• The de Rham theorem for general spaces. 1966