Is topology just a generalization of real analysis? 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 $\mathbb{R}$?
 A: 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.

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*Abstract the distance formula in $\mathbb R^n$ to a general metric. From this we go from real euclidean spaces to metric spaces.

*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.

*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.

A: 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.
A: 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 $\mathbb{R}^n$.
