# Explaining topology to non-mathematicians

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

HallsofIvy
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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).

matt grime
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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).

Galileo
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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.

matt grime
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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.

Icebreaker
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?"

robphy
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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

arildno
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Dearly Missed
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...)

robphy
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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.

Haelfix
The problem I have with explaining topology is it often sounds like im 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 im 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.

HallsofIvy
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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. 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.

George Jones
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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?

matt grime
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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?

EnumaElish
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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?

EnumaElish