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This question might seem philosophical a bit:

in a standard manifolds introductory course. when one talks about open , closed sets in $\mathbb{R}^n$ it's always the standard euclidian topology that we think about. Is there a more general framework in which one defines manifolds in different topologies in a way that generalize the standard setting that is closer to the physical world of course.

Thank you in advance for your answers

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I'm not sure what you're asking. Are you asking whether there are things like manifolds, but where the local model is not open subsets of $\mathbb{R}^n$ but something else? I also don't know what you mean by "closer to the physical world." Manifolds are close enough for general relativity... –  Qiaochu Yuan May 15 '11 at 3:30
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I mean having a manifold in a general space X with a random topology and say something like each point of the manifold is in an open U of X that is homeomorphic to an open of an other space Y equipped with an other topology. –  El Moro May 15 '11 at 3:35
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I'm not sure what you're asking. The main idea of the notion of a smooth manifold is that you model an abstract topological space locally on some other space in which the notion of differentiation makes sense. The easiest way to do this is to use $\mathbb{R}^{n}$ or $\mathbb{C}^{n}$ (for complex manifolds) as a model. There are also infinite-dimensional versions, modeling manifolds on Hilbert spaces, Banach spaces or Fréchet spaces (used in particular for the investigation of spaces of smooth maps). –  t.b. May 15 '11 at 3:40
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@El Moro: okay, yes, such things exist, but I still don't know what you mean by "closer to the physical world." –  Qiaochu Yuan May 15 '11 at 4:13
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@El Moro: okay, in that case, 1) please clarify the question in the OP, as it is very confusing, and 2) why did you accept the answer below? The answer is yes, we absolutely can do that, but the answer below has nothing to do with that question. –  Qiaochu Yuan May 15 '11 at 6:17
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Wrapping up the comments under the original question: One viewpoint of (differentiable) manifolds is that manifolds are those objects that can be approximated locally by a linear space, the "model space", so that concepts of calculus can be transported from the model space to the manifold.

The model space needs to be at least a topological vector space, a vector space that is also a topological space such that the algebraic operations are continuous, to make sense of notions of calculus. You first need to be able to define e.g. the derivative of a path in your model space before you can define the derivative of a path in the manifold.

If you take a curve in your model space $E$:

$$ c: \mathbb{R} \to E $$

you'd need to be able to write down the definition of the derivative $c'$:

$$ c'(t) := \lim_{s \to 0} \frac{1}{s} (c(t+s) - c(t)) $$

In order to make sense of this formula you need the linear structure, a topology, and the continuity of addition and scalar multiplication.

The simplest example of a topological vector space is of course $\mathbb{R}^n$. It is a theorem that on a finite dimensional vector space there is one and only one topology that turns the vector space into a topological vector space, which in the case of $\mathbb{R}^n$ is the canoncial topology induced by, for example, the Euclidean norm.

Therefore, when one talks about differentiable manifolds modelled on $\mathbb{R}^n$, $\mathbb{R}^n$ is always equipped with this unique topology. The same applies to complex manifolds modelled on $\mathbb{C}^n$, too, of course. Things get more interesting for model spaces that are infinite dimensional, like Fréchet spaces. But as long as you are talking about differentiable manifolds modelled on finite dimensional spaces, the topology is fixed by the requirement that the model space needs to be a topological vector space.

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The definition of a manifold is reasonably nice topological space that locally looks like Euclidean space. More formally, $M$ is a topological manifold of dimension $n$ if $M$ is second countable and Hausdorff and for each $p \in M$ there exists an open set $U \ni p$ (in whatever topology $M$ has) such that there is a homeomorphism $U \stackrel{\approx}{\to} V$, where $V \subset \mathbb R^n$, open.

It sounds like you've either only seen manifolds as subsets of Euclidean space or you're wondering if we can replace the homeomorphism with subsets of $\mathbb R^n$, with the usual topology, with subsets of $\mathbb R^n$ with a different topology. In either case, hopefully understanding the formal definition helps. We only call something a manifold if it locally looks like $\mathbb R^n$ with the usual topology.

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Algebraic geometry essentially uses the same construction to define varieties — I won't go into details, but basically we develop the theory for the special case of affine varieties, then build more general algebraic varieties by gluing affine varieties together along open sets. The topology (the Zariski topology) used here is markedly different — for example, it's rarely ever Hausdorff because the open sets are far too large.

However, in a sense, it's not really a generalisation of the concept of manifold — we still have a canonical class of spaces which we glue together, and this class of spaces is fixed in advance for the whole subject. Moreover, although there is a way to make some varieties into manifolds, not every variety is a manifold, and not every manifold is a variety.

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The topology for $\mathbb{R}^{n}$ is very nice. If you try to replace this topological space by some random topological space $X$ you will have to come to terms with whatever goofy properties the topology for $X$ has. Suppose we call this new type of object $M$ an $X$-manifold. There seems to be two paths to proceed

  1. Show that if $X$ has some nice property then some statement about $M$ is true.
  2. Define all properties of $M$ in terms of the properties of a general topological space.

In the first case you end up butterfly-collecting. For each nice property you might get a theorem. In the second case the results you get will upon your ability to define manifold-like properties in terms that make sense for an arbitrary topological space. An example of the latter case might be the transition from $\delta$ and $\epsilon$ proofs to point set topology or locales. In the first transition one eliminates the metric; in the second transition one eliminates the use of points. In certain instances the reintroduction of a metric or points, respectively, may prove beneficial.

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