How is topology and real analysis applied in economics?

Brian K: How is topology and real analysis applied in economics?
How are concepts like compactness and the topology of Euclidean spaces even applicable to economics? I can understand how calculus, linear algebra, and differential equations are useful for economics, but topology and real analysis seem quite abstract to have any useful application.

Furthermore, can someone comment on whether algebraic topology or Von Neumman algebras can be useful for economic study as well?

Thanks!

Answers and Views:

Answer by Bored Goblin
Isn’t calculus part of real analysis?

General results might still be used to prove various theorems. And once you get to graduate-level economics, many problems are functional equations, and while they sometime reduce to simple differential equations, many other times you need general functional analysis to prove things.

Answer by I didn’t do it!
I have yet to see an economic problem which would need to make use of abstract concepts rooted algebraic or geometric topology. Of course, the mathematical properties of Euclidean spaces apply also to economic models if they make use of such spaces. For example, the Bolzano–Weierstrass theorem as a particular case of compactness in a finite dimensional Euclidean space would apply to a representation of a multidimensional utility function in the Euclidean space, given the right properties, as much as it would apply to any other scientific application. However, it has hardly any relevance for the outcome that an economist could derive from such a functional.

Bottom line is that mathematics should only be considered as a tool or a language used by economists to articulate relationships between economic variables. However, when the mathematical properties of the mathematical models become more important than the underlying economic relationships, there is a significant risk that such models lead to wrong conclusions. It seems that in recent years many economists, in particular in financial economics, attempted to elevate the economic science to the same level as, let’s say quantum physics, with the deplorable result that the actual economic concepts got lost in abstract discussions of the properties the tools used to describe the underlying concept.

Some further thoughts on this subject:
https://krugman.blogs.nytimes.com/2011/02/02/models-plain-and-fancy/