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smooth_atlas [2014/12/18 18:13] nikolaj |
smooth_atlas [2014/12/18 18:44] nikolaj |
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| ===== Smooth atlas ===== | ===== Smooth atlas ===== | ||
| ==== Set ==== | ==== Set ==== | ||
| + | |||
| | @#55CCEE: context | @#55CCEE: $\langle M,T\rangle$ ... second-countable Hausdorff space | | | @#55CCEE: context | @#55CCEE: $\langle M,T\rangle$ ... second-countable Hausdorff space | | ||
| | @#55CCEE: context | @#55CCEE: $n\in \mathbb N$ | | | @#55CCEE: context | @#55CCEE: $n\in \mathbb N$ | | ||
| Line 8: | Line 9: | ||
| | @#55EE55: postulate | @#55EE55: $\phi\circ\psi^{-1}$ ... smooth | | | @#55EE55: postulate | @#55EE55: $\phi\circ\psi^{-1}$ ... smooth | | ||
| - | === Formalities === | ||
| A priori "$\phi\circ\psi^{-1}$" in the postulate doesn't make sense as their domains/codomains will not much. Here, really, one must choose functions with appropriately restricted domain. | A priori "$\phi\circ\psi^{-1}$" in the postulate doesn't make sense as their domains/codomains will not much. Here, really, one must choose functions with appropriately restricted domain. | ||
| - | ==== Discussion ==== | + | === Universal property === |
| + | For a smooth atlas there are smooth coordinate changes on $\mathbb R^n$. | ||
| + | |||
| + | ----- | ||
| === Idea === | === Idea === | ||
| For a smooth atlas there are smooth coordinate changes on $\mathbb R^n$. | For a smooth atlas there are smooth coordinate changes on $\mathbb R^n$. | ||
| - | ==== Parents ==== | + | ----- |
| === Context* === | === Context* === | ||
| [[Second-countable Hausdorff space]], [[Natural number]] | [[Second-countable Hausdorff space]], [[Natural number]] | ||
