# Explaining topology to non-mathematicians

• octol
In summary, topology is the study of limits and local properties. It is the most general object in which you can do calculus and includes all limits, including continuity. Topology is important because it helps connect many branches of mathematics and is the basis for understanding smooth changes and continuous functions. It has been used to restate and reinvigorate parts of mathematics and has applications in physics, such as in understanding black holes. Topology is often misunderstood as being related to topography, but it is actually a fundamental concept in mathematics.
octol
I've been trying to explain what topology is, and why it is important, to non-mathematicians. Specifically to other (non-theoretical) physicists. To best explanation I can come up with is along the lines of "generalized geometry one step up from set theory", and that it is important because it "connects many branches of mathematics".

What do you all think about that?

Topology gives the conditions for the 99 % of the theorems in Calculus.

well i don't know about 99% of the theorems in calculus; differentiation & integration aren't topological concepts. topology is very important though. wikipedia's elementary intro to topology is pretty good i think:
http://en.wikipedia.org/wiki/Topology

good ol mathworld has a page about it also, but it's got some technical math talk on it
http://mathworld.wolfram.com/Topology.html

Topology is basically the study of limits. That's why differentiation and integration are not topics in topology- for those you need to also be able to do arithmetic. The most general object in which you can do calculus is a topological vector field.

All limits, including continuity are included in topology. A topological space is the most general object in which you have limits. For people who have not studied calculus and are not clear on what limits themselves are, you might fall back on "rubber-sheet geometry"- but I would emphasize that the reason we want to study such things is because we are interested in how far we can go with "smooth" changes- no breaks in the sheet (i.e. continuous changes).

Topology is, as has been said, the most elementary (that is with minmal hypotheses, rather than simplest) way to talk of continuity and the study of "local" properties. Local means that we define things on small patches (the open subsets). Continuity is a purely local idea. We may then say that topolgoy is the attempt to patch together these local bits of datum. How these things patch together (if at all) tells us about the underlying shape of the space we care about. we can for instance on the surface of the sphere define on small patches a smooth vector field, that is a smoothly varying choice of tangent vector at each point, but owing to topological properties we cannot piece these together to get one defiend on all the sphere (globally). THis is commonly stated as "there is no way to comb the hair on a tennis ball so it is flat everywhere". This has ramifications in, say, fluids where we can think of this as saying there is no smooth flow of fluid on the surface of the sphere, ie there must be a sink and a source (or both simultaneously).

This language of topology has been used to restate and reinvigorate lots of parts of mathematics (mathematics essentially being an exercise in cross pollination).

I tried telling my firends about what topology was: How we define spaces without the concept of distance and that a cube and a pyramid and a sphere are all the same from a topological point of view.

In the end they declared I was nuts.

Well, they might well do if you don't explain that often spaces are only important upto homeomorphism. How you'd get that across is a bit of a tricky one.

The first thing everyone to whom I talked about think of is topography. God, that's annoying. "You want to study maps for a living?"

One can discuss topology at various levels.

Here is a description of a course "TOPOLOGY for undergraduates and scientists"
http://www.math.purdue.edu/~gottlieb/Courses/top490598
which has some useful motivation.

http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Topology_in_mathematics.html gives some of the history.

Here is an essay which highlights some important uses of topology and physics
http://arxiv.org/abs/hep-th/9709135 . You can probably get some ideas from here, although you'll probably have to simplify it for your audience.

This describes point-set topology applied to black holes at an elementary level
http://www.grc.nasa.gov/WWW/K-12/Numbers/Math/Mathematical_Thinking/blackhl.htm
http://www.grc.nasa.gov/WWW/K-12/Numbers/Math/Mathematical_Thinking/blackhl_two.htm

Just a thought:
I think that one should start at the "local" level, and make the non-mathematician appreciate the crucial importance the property of continuity (and "similar" properties) has in enabling us to glean information of the local surroundings about a point given exact info about the point.
In this hand-wavy manner, it would (perhaps) seem natural that the next step is to deduce global properties through patching together the info you've collected on various local levels.

HallsofIvy said:
Topology is basically the study of limits. That's why differentiation and integration are not topics in topology- for those you need to also be able to do arithmetic. The most general object in which you can do calculus is a topological vector field.

I had been wondering about that. It has been pointed out in other threads that you don't need an inner product to do calculus (as in Riemannian geometry). But do you necessarily need a norm? Can you do calculus with just a metric defined on the vector space? I haven't thought about it yet, but i guess it could be that the structure imposed by the vector space axioms would require that any metric is automatically a norm too.

PhilG said:
I had been wondering about that. It has been pointed out in other threads that you don't need an inner product to do calculus (as in Riemannian geometry). But do you necessarily need a norm? Can you do calculus with just a metric defined on the vector space? I haven't thought about it yet, but i guess it could be that the structure imposed by the vector space axioms would require that any metric is automatically a norm too.
other way around, the norm is automatically a metric if you write d(x,y) = ||x-y||

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fourier jr said:
other way around, the norm is automatically a metric if you write d(x,y) = ||x-y||
Right. I am asking if you can do calculus with JUST a metric, rather than a norm.

HallsofIvy said:
Topology is basically the study of limits. That's why differentiation and integration are not topics in topology- for those you need to also be able to do arithmetic. The most general object in which you can do calculus is a topological vector field.

All limits, including continuity are included in topology. A topological space is the most general object in which you have limits. For people who have not studied calculus and are not clear on what limits themselves are, you might fall back on "rubber-sheet geometry"- but I would emphasize that the reason we want to study such things is because we are interested in how far we can go with "smooth" changes- no breaks in the sheet (i.e. continuous changes).

more like analysis is the study of limits. i don't see the word limit anywhere in any toplogy text, not even in the stuff about nets & filters, which are generalizations of the sequence concept. to have limits you need a metric & to have a metric you need a metrizable space, but not every space is metrizable. the topological definition of continuity also doesn't use any limits; it says that the preimage of an open neighbourhood of f(x_0) in the target space is an open neighbourhood of x_0 & that's all.

PhilG said:
Right. I am asking if you can do calculus with JUST a metric, rather than a norm.
not sure what you mean; every first-year calculus person does calculus with a metric since they're dealing with distances between points so yes you can if that's what you mean. or maybe you mean is it possible to do calculus with a metric that isn't a norm. in that case i would say probably (haven't really thought about it) but i don't think that would be useful in real life since the usual metric is the one used in calculus. (could be wrong about that...)

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fourier jr said:
more like analysis is the study of limits. i don't see the word limit anywhere in any toplogy text, not even in the stuff about nets & filters, which are generalizations of the sequence concept. to have limits you need a metric & to have a metric you need a metrizable space, but not every space is metrizable.

For topological spaces X and Y, a function f : X->Y, and points x, x0 in X and y in Y:

lim f(x) = y as x->x0 if for every neighborhood V of y, there is a neighborhood U of x0 such that f(U - {x0}) is contained in V. You don't need a metric for that.

fourier jr said:
or maybe you mean is it possible to do calculus with a metric that isn't a norm.
That's what I meant.

The problem I have with explaining topology is it often sounds like I am merely explaining manifolds and differential geometry. I try to emphasize that those are merely subsets of the possible *structures* and *shapes* that topology can make us think about. But then it starts getting hard to come up with examples, especially physical examples. The first thing I usually say to that, is that its very common to think of spaces that have no good notion of what length is, its harder though to explain further into what 'structure' these things can posses in the actual set theory portion of the axioms.

In fact, since I am a physicist now, its damned hard for me to think of things outside the realm of the 'standard topology', in fact I think its still an open question in many facets of physics (like say GR and quantum gravity). In many ways its a testament to the breadth and generality of Topology, but then again its also the biggest weakness.

fourier jr said:
more like analysis is the study of limits. i don't see the word limit anywhere in any toplogy text, not even in the stuff about nets & filters, which are generalizations of the sequence concept. to have limits you need a metric & to have a metric you need a metrizable space, but not every space is metrizable. the topological definition of continuity also doesn't use any limits; it says that the preimage of an open neighbourhood of f(x_0) in the target space is an open neighbourhood of x_0 & that's all.

No, analysis includes a lot more than limits. Analysis is basically the theory behind calculus- and, as I said before, you need at least a topological vector space for that.

I certainly have seen limits in topology texts: Suppose {an} is an infinite sequence of points in a topological space. Then an-> n (as n goes to infinity) if and only if, for any open set containing a, there exist N such that if n> N then an is also in that open set.
Same thing with limits of functions. If f: M-> N is a function from topological space M to topological space N, x0 is in M, then limit f(x)= b (as x->x0), b in N, if and only if for every open set V containing b, there exist an open set U containing x0 such that f(U) is a subset of V.
That, of course, would give the same definition of "continuous function" as the one you give.

Topology was largely ignored by mainstream physicists until 1965, when Roger Penrose used it in a short note to Physical Review Letters that outlined the first proof of a singularity theorem in general relativity. Before 1965, numerical studies had shown that spherical stellar collapse resulted in spacetime singularities. The singularities were thought to be the result of the spherically symmetric models used, but Penrose showed that stellar collapse under quite general physical conditions always produces a singularity if the collapse has proceeded far enough. Penrose's theorem is outlined in the arxiv link given by robphy in this thread.

Penrose's seminal work led to much research in general relativity using topological methods by Penrose, Hawking, Geroch, and others. Now, topology is used in many branches of physics. One of my favourite courses that I took as a student was a point-set topology course based on the book by Munkres.

In the popular-level "The Edge of Infinity: Beyond the Black Hole," Paul Davies explains some of these topological ideas remarkably well. Where Hausdorff topological spaces are involved, one's intuition usually is quite good. This book exploits this intuition to give a tremendous (and quite accurate) explanation of Penrose's first singularity theorem. I even recommend it as a complement to Wald and Hawking and Ellis for physics types trying to learn the technical details of singularity theorems.

A simple example from general relativity: Using topological methods, it is easy to show that any compact spacetime admits closed timelike curves.

Regards,
George

HallsofIvy said:
If f: M-> N is a function from topological space M to topological space N, x0 is in M, then limit f(x)= b (as x->x0), b in N, if and only if for every open set V containing b, there exist an open set U containing x0 such that f(U) is a subset of V.

f(x0) is not necessarily in V. Also, is a topological vector space really all the structure you need to do calculus? Don't you at least need a metric? For example, can you do calculus in a vector space in which every set is an open set?

a function is continuous if and only if the inverse image of an open set is open. there is no need to mention metrics or norms at all, it is purely a topological definition.

matt:

So the answer to my question
Also, is a topological vector space really all the structure you need to do calculus?
is yes?

PhilG said:
Right. I am asking if you can do calculus with JUST a metric, rather than a norm.
IMHO another way to put the question is "can you define limits in a metric space without a norm"? If you can define limits, then you can define derivative and integral, am I not right?

matt grime said:
a function is continuous if and only if the inverse image of an open set is open. there is no need to mention metrics or norms at all, it is purely a topological definition.

EnumaElish said:
IMHO another way to put the question is "can you define limits in a metric space without a norm"? If you can define limits, then you can define derivative and integral, am I not right?

Limits are defined for general topological spaces, even ones that don't have a metric. I'm not sure how to define a derivative without a norm though. Suppose you have a function f:Rn-->Rm It seems as though you need a way to associate a number with each displacement vector h in Rn, i.e. a norm. The usual way

$$f'(a) = \lim_{h \rightarrow 0} \frac{f(a+h) - f(a)}{h}$$

doesn't make sense because you can't divide a vector in Rm by a vector in Rn, or any other vector for that matter. I just looked at Spivak's "Calculus on Manifolds", and he uses a normed version

$$\lim_{|h| \rightarrow 0} \frac{|f(a+h) - f(a) - \lambda(h)|}{|h|} = 0$$

to define the derivative of f at a as the unique linear map λ : Rn-->Rm that satisfies the above equation. So I guess a better way to ask my question is: Is there a way to define the derivative of a function from Rn to Rm that doesn't require the use of a norm?

HallsofIvy said:
No, analysis includes a lot more than limits. Analysis is basically the theory behind calculus- and, as I said before, you need at least a topological vector space for that.

yeah but limits are the fundamentals in a calculus or analysis course. you have to know that stuff to be able to do other calculus/analysis stuff like integration & differentiation, sups & infs, etc.

EnumaElish said:
no, derivative is defined as a limit, at least in calculus

re: limits, I've seen topics like convergence in topology books, but not the limit of a sequence or function, except maybe a 1/2-page digression at the end of a section. contrast the amount of discussion involving limits in a topology text & in an analysis/calculus text. nothing is defined or stated in terms of limits in topology either; it's all about point-sets & how they relate to each other.

in R^n f is differentaible at x if there is a linaer map Df(x) satisfying

f(x+h)=f(x)+Df(x)h + ho(|h|)

this can be extended to any place where there is a notion of linear map or a map such as |?| to the reals, or some other ordered space.

then there are formal derivatives: it is perfectly possible to define the derivative of X^2 to be 2X where X^2 is thought of as a polynomial over any ring.

we also have derivations too defined on any ring.

but whether or not you can define derivatives isn't important. if you like derivatives are something particular about R or C (or vector spaces over them) where we can interpret them geometrically. They are somehow more complicated than limits since they require you to be able to add things up, f(x+h), and talk about |h|, or in the more traditional one dimensional version, to divide by h. Thus while they use limits they also use seomthing more of the properties of the udnerlying space. it is possible to do these things such as derivative and integral in more abstract setting but it is not necessarily useful to discuss here.

matt grime said:
in R^n f is differentaible at x if there is a linaer map Df(x) satisfying

f(x+h)=f(x)+Df(x)h + ho(|h|)

this can be extended to any place where there is a notion of linear map or a map such as |?| to the reals, or some other ordered space.

then there are formal derivatives: it is perfectly possible to define the derivative of X^2 to be 2X where X^2 is thought of as a polynomial over any ring.

we also have derivations too defined on any ring.

but whether or not you can define derivatives isn't important. if you like derivatives are something particular about R or C (or vector spaces over them) where we can interpret them geometrically. They are somehow more complicated than limits since they require you to be able to add things up, f(x+h), and talk about |h|, or in the more traditional one dimensional version, to divide by h. Thus while they use limits they also use seomthing more of the properties of the udnerlying space. it is possible to do these things such as derivative and integral in more abstract setting but it is not necessarily useful to discuss here.

please can you tell me how ? and what you mean by polynomial over a ring ?

and matt, is the example you gave of hair on the tennis ball, similar to the Earth being flat locally ?

do you know what a ring is? let R be a (commutative) ring with unit and let R[x] be the set of all polynomials in x with coefficients in R. we can differentiate any element of R[x] formally, ie without reference to limits. D(ax^2)=2ax this is well defined since R has a unit, 1, and 2=1+1. the usual rules follow: D(pq)=pD(q)+D(p)q and D(p+q)=D(p)+D(q) etc. Example R= integers mod 2, then D(x^2)=0, and this is a very important algebraic fact as anyone who's done an intro to galois theory will remember, though if yuo have a memory like mine you won't recall exactly what the property is though separability seems to be a good bet.

as for your second question my answer is: eh? don't understand what you want to know. what does "similar" mean in this context? they are both things to do with tangent spaces i suppose, and how the local tangent spaces glue together globally ie although we can on each small area of the surface of the tennis ball comb the hair in the same direction since locally the tennis ball loooks a lot like a flat object, and on a small flat disc we can certainly, at each point, pick a direction vector and let them vary smoothly, eg if we think of it as a disc in the xy plane just have the vector pointing in the x direction with constant length 1 at each point, but we cannot take these local things and patch them together to get a globally smooth one. hopefully that answers the question.

matt,

but what's the advantage of defining derivatives without reference to limits ?

because it applies to non-normed spaces and it has simply proved useful. if it hadn't then it would have been forgotten. example. suppose f is a poly in k[x] and it splits over k, then if f has a repeated root then f and f' are not coprime. this is a result of use in galois theory, for example. we also care about tangent spaces for abstract algebraic varieties too (lie algebras).

matt grime said:
in R^n f is differentaible at x if there is a linaer map Df(x) satisfying

f(x+h)=f(x)+Df(x)h + ho(|h|)

this can be extended to any place where there is a notion of linear map or a map such as |?| to the reals, or some other ordered space.

Is that last term "the vector h, scaled by a number that goes to zero faster than |h|"? What if h and f(h) live in spaces of different dimensions, say n and m?

That's a typo, the general Taylor's polinomium with Peano's remainder is:

$$f(\vec{x_0}+\vec{h})= \sum_{k=0}^n \frac {{d^k}f|_{\vec{x_0}}(\vec{h})} {k!} + o(|\vec{h}|^n)$$
for $$\vec{h} \rightarrow \vec{0}$$

where differentials should be considered as functions $${d^k}f:\mathbb{R}^n \rightarrow \mathbb{R}$$ of the vector $$\vec{h}$$ evaluated at the point $$\vec{x_0}$$.

Saying: $$g(x)=o(f(x))$$ for $$x \rightarrow x_0$$ means that
$$\lim_{x\rightarrow x_0} \frac {g(x)} {f(x)} = 0$$

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