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smooth_atlas [2014/12/04 16:42] nikolaj |
smooth_atlas [2014/12/18 18:44] (current) 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: $n\in \mathbb N$ | | ||
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| @#55EE55: postulate | @#55EE55: $\phi\circ\psi^{-1}$ ... smooth | | | @#55EE55: postulate | @#55EE55: $\phi\circ\psi^{-1}$ ... smooth | | ||
- | ==== Discussion ==== | ||
- | === 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. | ||
+ | |||
+ | === 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]] |