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While trying to learn undergraduate topology, I came across this lecture by Dr. Zimmerman who claims "Topology is a generalization of real analysis, a lot of topology anyway." They are obviously related and topology does seem more general, but this statement still surprised me.

Can all of real (and complex) analysis be recast in the framework of topology?

Edit: Could I say that real analysis is just studying the topology of $\mathbb{R}$?

Mithrandir
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  • In real analysis you just consider a metric space. – Wuestenfux Aug 26 '20 at 17:23
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    Real analysis includes calculus. Topology, as such, does not. – herb steinberg Aug 26 '20 at 17:26
  • I agree with @Lt.Commander.Data - in real analysis you are really only studying $one$ topology, that is, the topology of $\mathbb{R}$. While in a topology course, you study an array of different topologies. – Taylor Rendon Aug 26 '20 at 17:32
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    The spirit of the remark is that a topology is a collection of open neighborhoods (which are in some sense telling you points you are allowed to wiggle a little and get to), and in analysis these neighborhoods are usually dictated by a metric. – Ted Shifrin Aug 26 '20 at 17:35
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    Side comment: It's best not to delve on such trivialities, especially since both are very wide fields. The two are heavily linked, yes, but are different enough that the disjoint elements are probably greater in number than the overlap. The question at the end, though, is interesting. – Lt. Commander. Data Aug 26 '20 at 17:38
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    You can view topology as an abstraction of geometry. – Simon Marynissen Aug 26 '20 at 17:39
  • Even though topology doesn't really cover integration theory, I do like to think of the Riemann integral as the limit of a net where the index set is partitions of the interval ordered by refinement. – Daniel Schepler Aug 26 '20 at 19:23
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    See https://math.stackexchange.com/questions/272278/is-there-a-topological-notion-of-the-derivative and https://math.stackexchange.com/questions/1267268/why-cant-differentiability-be-generalized-as-nicely-as-continuity; Differentiation doesn't generalize as well to topological spaces as ideas of continuity and limits do. – Alex Jones Aug 27 '20 at 16:14

3 Answers3

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The structure of $\mathbb R$ is richer than its (usual) topology. Its topological properties would make it "metrisable" but would not provide it with a metric, and you need a metric to do worthwhile analysis. The (usual) metric on $\mathbb R$ is usefully compatible with its algebraic properties.

There are topologies on $\mathbb R$ other than the usual one, and alternative metrics also exist. So analysis in the usual sense cannot be reduced in a simple way to topology. The particular useful structures on $\mathbb R$ are particular, and useful.

One (helpful) way of looking at topology is as trying to capture the essence of continuity (which can be expressed in terms of open sets). This is obviously important in analysis, but isn't the whole story.

Mark Bennet
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Yes. In fact, the morphisms of a topological space are continuous functions.

To proceed from real analysis to topology we must do a few things.

  1. Abstract the distance formula in $\mathbb R^n$ to a general metric. From this we go from real euclidean spaces to metric spaces.
  2. Using the metric space definition of continuity, we prove that a function is continuous if and only the inverse of an open function is open. This theorem eliminates the need for the metric in the definition of continuity.
  3. We then use the closure properties of concrete open sets in metric spaces to define a topology and from that we get a topological space.
John Douma
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    But the morphisms in Euclidean spaces are differentiable (or even smooth). – Qi Zhu Aug 27 '20 at 07:15
  • $\mathbb R^n$ can be given many structures. That is not really relevant to the statement "Topology is a generalization of real analysis, a lot of topology anyway." – John Douma Aug 27 '20 at 17:50
  • Of course, it can be given many structures but real analysis is the study of the category of Euclidean spaces - the main goal is to study how differentiable functions behave, not merely continuous functions. – Qi Zhu Aug 27 '20 at 17:53
  • @QiZhu Right, but we can abstract those concepts away and generalize to get new structures. Do you believe metric spaces are not a generalization of Euclidean spaces because metric spaces don't have the concept of differentiability? – John Douma Aug 27 '20 at 17:59
  • Yes, those are completely different categorical structures - metric spaces don‘t have a notion of differentiability as you say. There is no forgetful functor (as one would otherwise expect) from the category of metric spaces to the category of Euclidean spaces. This is not a matter of abstraction, metric spaces see different things than the category of Euclidean spaces (they do not see the concepts related to differentiability). – Qi Zhu Aug 27 '20 at 18:21
  • Almost the same example is whether topology is a generalization of differential geometry - smooth manifolds hold a topology after all. (This is almost the same example because the category of Euclidean spaces of course embeds into the category of manifolds.) Similarly, they are simply different categories that behave completely differently. – Qi Zhu Aug 27 '20 at 18:24
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It depends on what you mean by "generalization".

I wouldn't say topology is a complete generalization of real analysis since in real analysis we often explicitly use the algebraic structure of the field $\mathbb{R}$. For instance, the fact that for each $x \in \mathbb{R}$, there exists a multiplicative inverse $x^{-1}$ is a trivial fact used ubiquitously in analysis, but it has nothing to do with $\mathbb{R}$'s topological structure.

But it is a generalization in a more looser sense. Topology abstracts the notion of "distance" and "closeness" in $\mathbb{R}^n$.

MathematicsStudent1122
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  • Naturally, I feel the distance is an algebraic order concept (in the sense $d(a,b)=b-a$, where $a \leq b$), right? Can you specify how the concept of 'distance' fits a topological discription? Of course, I can see the generalization of 'open balls' as 'base' for a topology. But I cannot directly connect distance within a generic base for a topology. Thanks in advance. – Messi Lio Nov 07 '22 at 05:18