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In page 7. of Fecko's book on Differential Geometry and Lie Groups, he says,

In an effort to map a bigger part of a country, an atlas (a collection of maps) has proved to be helpful. A good atlas should be consistent at all overlaps: if some part of the land happens to be on two (or more) maps (close to the margins, as a rule), information obtained from them must not be mutually contradictory.

How do you construct charts that are mutually contradictory? The only example I can think of is when one map fails to become a bijection, but this is hardly an admissible chart.

Ron
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    Here's an example for you. Suppose your country is one-dimensional. Suppose cities $A$, $B$, $C$, and $D$ appear in both of your maps. Would it be possible for them to be in the order $A,B,C,D$ on one map and in the order $A,C,B,D$ on the other? – Ted Shifrin Jul 01 '17 at 16:20
  • Yes, for instance $x \mapsto x$ and $x \mapsto -x$ are homeomorphisms. But I'm still unsure how this is a contradiction. (Actually, I don't even know what a contradiction should be in this context... this is because I'm not sure if I should think about ordering in a topological space, when there might not be such a structure.) – Ron Jul 01 '17 at 16:38
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    Well, I restricted to a one-dimensional manifold. Any connected one-dimensional manifold is either an open interval or a circle. If we were in a two-dimensional manifold, we could have a diffeomorphism that fixes two points of the four and switches the other two. – Ted Shifrin Jul 01 '17 at 16:40
  • I see now, thank you! – Ron Jul 01 '17 at 16:54
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    The type of phenomenon Fecko has in mind is using the homeomorphisms $x \mapsto x^{1/3}$, $x \mapsto x$, and $x \mapsto x^{3}$ as charts for a one-dimensional universe; motion that looks "regular" (differentiable, with non-zero velocity) in one chart may not look regular in another. (In a similar vein but lying much deeper, in dimension four there exist pairs of compact smooth manifolds that are homeomorphic but not diffeomorphic.) – Andrew D. Hwang Jul 01 '17 at 16:54

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