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Question came up when reading this answer. How is it possible that we can choose different topological spaces to model a same physical scenario?

If we have such different spaces, so many things will be different. For example, convergent sequences, what points there can be and qualitative features like holes. So wouldn't these features mess up our model?

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    $\begingroup$ Diffeomorpjisms is not exactly about coordinate systems. The shift in coordinates are jn the chart transition maps not in diff manifold level @FlatterMann $\endgroup$
    – Babu
    Oct 25, 2022 at 6:28
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    $\begingroup$ It is still correct that physics is not the same as the description of physics. The physics of a system is unique, but there is an infinity of different descriptions for it. That is a redundancy that only exists in the mathematics. On the physical level two cars either collide in the intersection or they don't. How you describe that does not matter. $\endgroup$ Oct 25, 2022 at 6:38
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    $\begingroup$ @TrystwithFreedom sorry I do not get the question. The answer you linked just says that you can study gravity (GR or Newton gravity) over different base manifolds.. that's all. Similarly, you may study heat conduct on a sphere or on a torus, or the plane... What's the question? $\endgroup$
    – Quillo
    Oct 25, 2022 at 7:22
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    $\begingroup$ The choice of models effect s the features of space in consideration. So, are there no properties of motion related to feature of space? Is Motion completely independent of the arena it takes place in? Hope it makes more sense now @Quillo $\endgroup$
    – Babu
    Oct 25, 2022 at 7:23
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    $\begingroup$ @TrystwithFreedom now it's clear, thank you. Locally the equations of motion EOM are the same (for manifolds that are locally flat). But the EOM are not the only thing that define the evolution (boundary conditions and initial conditions are also needed, these depend on the global topology). It may be instructive to start with the heat equation on the infinite plane VS heat on a torus to get a feeling of the difference. Ofc different topologies do not refer to the same physical situation (it's different to live on a sphere rather than on a plane). $\endgroup$
    – Quillo
    Oct 25, 2022 at 7:36

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A (topological) manifold $(\mathcal M,\mathcal O)$ is locally homeomorphic to $\mathbb R^n$, which we implicitly take to be equipped with the standard topology. So already, local topological questions like topological completeness are answered in the affirmative.

The global topology of a manifold is another story - so if you mean holes in the sense of singular homology, then indeed you have an infinity of possible choices you could make (possibly subject to additional constraints you wish to impose for physical, mathematical, or philosophical reasons). Generically such features would have an observable effect, so it is then a matter of choosing the model which fits best with your observations.

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  • $\begingroup$ Oh okay, so the resolution is that locally speaking, all the topological manifolds are equivalent to each other. $\endgroup$
    – Babu
    Oct 25, 2022 at 7:44
  • $\begingroup$ physics.stackexchange.com/questions/54531/… $\endgroup$
    – Babu
    Oct 26, 2022 at 11:12
  • $\begingroup$ @TrystwithFreedom Do you have a follow-up question? $\endgroup$
    – J. Murray
    Oct 26, 2022 at 14:12
  • $\begingroup$ physics.stackexchange.com/questions/733848/… $\endgroup$
    – Babu
    Oct 26, 2022 at 14:12
  • $\begingroup$ @TrystwithFreedom I said that a manifold is locally homeomorphic to $\mathbb R^d$. That means that for each point $p$ we can find a small neighborhood of $p$ which is homeomorphic to a small patch of $\mathbb R^d$. It certainly does not imply that all $d$-dimensional manifolds are homeomorphic to a subset of $\mathbb R^d$. $\endgroup$
    – J. Murray
    Oct 26, 2022 at 14:19

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