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?
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.
