Definitions of topology and analysis

[Fredrik]

Definitions of "topology" and "analysis"

How do you define "topology" and "analysis"? I'm tempted to say that topology is the mathematics of...anything that involves limits. (Open and closed sets, continuous functions, etc...they can all be defined in terms of limits). But if that's an acceptable definition, how is "analysis" different? Are the two the same? Is one of them a proper subset of the other? Is the difference that topology is about the spaces and the points they contain, while analysis is about functions between those spaces?

 

[Pythagorean]

studying physical networks, we use topology as synonymous with "the way things are connected", as per graph theory.

Looking at the wiki, it appears this is exactly analogous to the more general topology study of manifolds. Graph theory simply deals with low dimensional systems, but as you increase the number of elements and connections to infinity (decreasing their size to infinitesimal) you can represent manifolds in n-dimensional space, but you are still essentially looking at how the space is "connected".

Topological spaces show up naturally in almost every branch of mathematics. This has made topology one of the great unifying ideas of mathematics.

The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are put together. For example, the square and the circle have many properties in common: they are both one dimensional objects (from a topological point of view) and both separate the plane into two parts, the part inside and the part outside."

i.e. "connectionism"

 

[HallsofIvy]

I would not say "topology is anything that involves limits" precisely because that would, as you say, include analysis. Rather, "topology is the study of limits". Now, if we add to that algebraic properties, being able to add and subtract elements, multiply and divide, so that we can, for example, form the difference quotient for a function and then take the limit we can define the "derivative' of a function and have analysis. In short, analysis is the combination of topology and algebra.

 

[disregardthat]

You should not ask how we define it, but how we can define it, for I see no sharp distinctions or definite rules for determining whether this-and-that is topology or really analysis, or the other way around (where do you stop doing analysis and start doing topology?). I think it is only a subtle but persuasive type of family resemblence which incline us to confidently categorize certain things as analysis, topology (or any other subject).

 

[micromass]

I think these can be nice definitions:

- Topology: study of spaces with a very loose notion of "closeness". This is enough to study things like continuity or compactness. Typically, one is more interested in extra structures like metric spaces or compact Hausdorff.

- Analysis: study of limits of functions. Typically, one is interested in nice enough functions: measurable, continuous, smooth, linear, functionals, etc.

 

[Fredrik]

You should not ask how we define it, but how we can define it, for I see no sharp distinctions or definite rules for determining whether this-and-that is topology or really analysis, or the other way around (where do you stop doing analysis and start doing topology?).

Yes, that's what I meant. I know that there are no "official" definitions of these terms. I'm just interested in finding out if it would be possible to write down exact definitions that would be acceptable to most mathematicians.

I had a similar discussion about linear algebra vs. functional analysis some time ago. I used to say that linear algebra is "the mathematics of finite-dimensional vector spaces and linear functions between them", and that functional analysis is "the mathematics of vector spaces and linear functions" (i.e. the same thing, minus the finite-dimensional requirement). I changed my mind as a result of that discussion. (If we prove e.g. that the smallest vector subspace that contains a given subset S is equal to the set of linear combinations of members of S, are we not doing linear algebra?) Now I prefer to say that a theorem about vector spaces and/or linear functions between vector space belongs to linear algebra if it doesn't involve topology, and to functional analysis if it does. (This doesn't fully define the terms, since it doesn't say which theorems belong to both, and which ones belong to neither).

I'm still not sure how to define those two terms. Right now I'm thinking that linear algebra is "the mathematics of vector spaces and linear functions", and that functional analysis is "the mathematics of topological vector spaces and linear functions between them". But that would make functional analysis a subset of linear algebra. That sounds weird to me, but maybe that's just because I used to think it was the other way round.

Maybe we should define functional analysis as "the mathematics of topological vector spaces and linear functions between them", and then define linear algebra by saying that "the mathematics of vector spaces and linear functions" is the disjoint union of linear algebra and functional analysis. (I don't think I like a definition that makes them disjoint, since many theorems in books on functional analysis books that are proved using methods from topology are straightforward generalizations of theorems from linear algebra).

Now, if we add to that algebraic properties, being able to add and subtract elements, multiply and divide, so that we can, for example, form the difference quotient for a function and then take the limit we can define the "derivative' of a function and have analysis. In short, analysis is the combination of topology and algebra.

Interesting point. I like it. But it seems to exclude some things that I think should be considered a part of analysis, like the basics of measure theory.

I think it would be accurate to say that almost all of analysis is about topological spaces that are also algebraic structures (topological vector spaces, topological groups, etc.), and the functions between them. But analysis is supposed to be a generalization of calculus, and some of the generalizations that mathematicians have made, have gone outside the context of topological spaces.

The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are put together.

I was told that topology can be described this way a couple of years before I even knew what a metric space was, but after having studied some topology, I don't think this description is accurate. It seems to be describing the concept of isomorphism classes in the category of topological spaces. This is certainly a part of topology, but I wouldn't say that this is what topology is.

I think these can be nice definitions:

- Topology: study of spaces with a very loose notion of "closeness". This is enough to study things like continuity or compactness. Typically, one is more interested in extra structures like metric spaces or compact Hausdorff.

- Analysis: study of limits of functions. Typically, one is interested in nice enough functions: measurable, continuous, smooth, linear, functionals, etc.

That looks nice, but I think this definition of analysis is too narrow. For example, don't we want the basics of measure theory to be a part of analysis? Your definition of topology says that it's the mathematics of topological spaces. Perhaps we should also say (as HallsofIvy did) that it's only considered topology if no algebra is involved? (Do we want the product rule for derivatives to be considered topology?)

 

[micromass]

Very interesting thread! Here are some of my thoughts:

I'm still not sure how to define those two terms. Right now I'm thinking that linear algebra is "the mathematics of vector spaces and linear functions", and that functional analysis is "the mathematics of topological vector spaces and linear functions between them". But that would make functional analysis a subset of linear algebra. That sounds weird to me, but maybe that's just because I used to think it was the other way round.

I think perhaps we can define

[tex]\{\text{functional analysis}\}=\{\text{linear algebra}\}\cap \{\text{topology}\}[/tex]

but even that definition is not complete, as there are still many things that are not covered by this. For example, Fredholm theory seems not to fit much in this definition.

Interesting point. I like it. But it seems to exclude some things that I think should be considered a part of analysis, like the basics of measure theory.

Controversial statement coming: I don't know if I would really classify measure theory as analysis. Obviously, integration is analysis, and obviously we need measure theory to perform analysis, but I don't think that it makes it part of analysis. Whenever I see measure theory, I think more about set theory and lattice theory than analysis.

I always consider measure theory to be a separate branch like topology. Maybe this is because I've first seen measure theory in the context of probability theory, then in the context of analysis...

I think it would be accurate to say that almost all of analysis is about topological spaces that are also algebraic structures (topological vector spaces, topological groups, etc.), and the functions between them. But analysis is supposed to be a generalization of calculus, and some of the generalizations that mathematicians have made, have gone outside the context of topological spaces.

I disagree that analysis is a generalization of calculus. I would argue that calculus is just "analysis dumbed down" (I hope nobody takes this the wrong way). In fact, I might even argue that calculus is no real part of mathematics at all (unless proofs are done, but then I consider it analysis).

That looks nice, but I think this definition of analysis is too narrow. For example, don't we want the basics of measure theory to be a part of analysis? Your definition of topology says that it's the mathematics of topological spaces. Perhaps we should also say (as HallsofIvy did) that it's only considered topology if no algebra is involved? (Do we want the product rule for derivatives to be considered topology?)

I don't know. Algebraic methods are used in topology, for example:

- the boolean algebra of sets is extensively used. For example, notions like filters, compactness, etc. all have generalizations in lattice theory. In my first class of topology, I always say that topology has more in common with algebra than with topology. Another controversial statement, but I do think it has some merit.

- I feel that things like topological groups are a part of topology. Statements like "every Hausdorff topological group is regular" are obviously topology.

- Algebraic topology uses a lot of algebra, of course. But I feel that you're talking about point-set topology, so I won't say anything about it. Still, things like [itex]\mathbb{R}^n[/itex] is homeomorphic with [itex]\mathbb{R}^m[/itex] iff n=m are obviously topological, but require deep algebraic methods.

A conclusionist remark: I feel that a definition of a certain mathematical branche can't really be given by saying what the branche studies. I think that we should make such a definitions by considering what techniques are used in the field.

For example, analysis uses techniques like "limits, linearization, function spaces, inequalities". While topology uses techniques like "limits, algebra of sets".
I think that looking at the techniques might yield more fruitful definitions than looking at the results...

Anyway, I find this a really cool discussion.

 

[Fredrik]

I think perhaps we can define

[tex]\{\text{functional analysis}\}=\{\text{linear algebra}\}\cap \{\text{topology}\}[/tex]

but even that definition is not complete, as there are still many things that are not covered by this. For example, Fredholm theory seems not to fit much in this definition.

There's also the fact that we need abstract algebra for at least a few important theorems in functional analysis.

Controversial statement coming: I don't know if I would really classify measure theory as analysis. Obviously, integration is analysis, and obviously we need measure theory to perform analysis, but I don't think that it makes it part of analysis.

I always consider measure theory to be a separate branch like topology.

Sounds reasonable.

I disagree that analysis is a generalization of calculus. I would argue that calculus is just "analysis dumbed down" (I hope nobody takes this the wrong way). In fact, I might even argue that calculus is no real part of mathematics at all (unless proofs are done, but then I consider it analysis).

But it's a pretty specific part of analysis that's dumbed down. Specifically the stuff about the topology of ℝ, functions from ℝ into ℝ, and derivatives and Riemann integrals of those functions. So why not call that part of analysis "calculus"? Hm, I guess the answer would be that the term "calculus" is older than the rigorous methods we use.

By the way, the term "calculus" doesn't really exist in Swedish. It's translated to "differential- och integralkalkyl", where "kalkyl" is a word that means "calculation" (or, I guess, "calculus") that is rarely used in any other context. The courses we were forced to take were called "Analys 1" and "Analys 2", and the optional course based on Rudin's "Principles..." was called "Analysens grunder" (foundations of analysis). They gave us a list of 35 theorems (maybe 45) that we were supposed to be able to state and prove for that first course, and there was a separate exam for the proofs.

I was quite surprised when I discovered (here at PF) that students from the USA consider "real analysis" an entirely different subject than "calculus".

topology has more in common with algebra than with topology

Are you sure about that?

- I feel that things like topological groups are a part of topology. Statements like "every Hausdorff topological group is regular" are obviously topology.

Agreed.

- Algebraic topology uses a lot of algebra, of course. But I feel that you're talking about point-set topology, so I won't say anything about it.

That's mainly because I know point-set topology pretty well and just barely have an idea what algebraic topology is. I got the impression that it's about using algebra to solve problems in topology. "If their fundamental groups are different, they can't be homeomorphic"...and stuff like that. I think this shows that my suggestion (that if it involves algebra, then maybe we shouldn't call it topology) is far too strong, but maybe there's something else we can say to ensure that derivatives aren't considered a part of topology.

A conclusionist remark: I feel that a definition of a certain mathematical branche can't really be given by saying what the branche studies. I think that we should make such a definitions by considering what techniques are used in the field.

For example, analysis uses techniques like "limits, linearization, function spaces, inequalities". While topology uses techniques like "limits, algebra of sets".
I think that looking at the techniques might yield more fruitful definitions than looking at the results...

Yes, that might be better.