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)

Last revised on June 16, 2020 at 13:31:30.
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