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smooth_manifold [2014/12/04 14:13] nikolaj |
smooth_manifold [2014/12/04 15:43] nikolaj |
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| | @#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: $n\in \mathbb N$ | | ||
| - | | @#FFBB00: definiendum | @#FFBB00: $\langle {\mathcal M},A\rangle\in$ it | | + | | @#FFBB00: definiendum | @#FFBB00: $\langle M,A\rangle\in$ it | |
| - | | @#55EE55: postulate | @#55EE55: $A$ maximal in atlas($\langle M,T\rangle,n$) | | + | | @#55EE55: postulate | @#55EE55: $A$ maximal in smooth atlas($\langle M,T\rangle,n$) | |
| ==== Discussion ==== | ==== Discussion ==== | ||
| === Elaboration === | === Elaboration === | ||
| - | Effectively, a smooth manifold would be given by providing //any// atlas. But then, due to the redundancy of some charts on small open sets, different atlases give rise to equivalent mathematical objects and so a smooth manifold is technically defined as the biggest and hence //unique// one amongst those objects. | + | Effectively, a smooth manifold would be given by providing //any// atlas. But then, due to the redundancy of some charts on small open sets, different atlases give rise to equivalent mathematical objects and so a smooth manifold is defined as the biggest and hence //unique// one amongst those objects. |
| === Reference ==== | === Reference ==== | ||
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| [[Second-countable Hausdorff space]] | [[Second-countable Hausdorff space]] | ||
| === Requirements === | === Requirements === | ||
| - | [[Atlas]], [[Maximal extension in a set]] | + | [[Smooth atlas]], [[Maximal extension in a set]] |
