Explaining topology to non-mathematicians

PhilG

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?

fourier jr

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:
Okay, how about derivative?
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.

matt grime

Homework Helper
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.

roger

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 ?

roger

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

matt grime

Homework Helper
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.

roger

matt,

but whats the advantage of defining derivatives without reference to limits ?

matt grime

Homework Helper
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).

PhilG

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?

Maxos

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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fourier jr

back to the beginning, topology is useful for proving that there are infinitely many primes!

consider the following topology on the integers. for $$a, b \in \mathbb{Z}, b>0$$ set $$N_{a,b}$$ = {$$a + nb | n \in \mathbb{Z}$$}.

Each set $$N_{a,b}$$ is a 2-way arithmetic sequence. a set G in this topology is open if either G is empty or if for every $$a \in G$$ there exists some b>0 with $$N_{a,b}$$ a subset of G. (not hard to check that unions & finite intersections are still open, and that Z & the empty set are all open so this makes a topology on the integers)

2 facts:
1) any non-empty open set is infinite
2) any $$N_{a,b}$$ is closed also. since $$N_{a,b} = \mathbb{Z}$$ \ $$\cup_{i=1}^{b-1} N_{a+i,b}$$ $$N_{a,b}$$ is the complement of an open set it's closed

now to use primeness. since any number except -1 or 1 has a prime divisor p & is therefore in $$N_{0,p}$$ we get that $$\mathbb{Z}$$ \ {-1,1} = $$\cup_{p\in\mathbb{P}}N_{0,p}$$

if the set of primes were finite, then $$\cup_{p\in\mathbb{P}}N_{0,p}$$ would be a finite union of closed sets & therefore closed. thus {-1,1} would be open, contradicting 1) above. thus there are infinitely many primes.

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Maxos

Very interesting but not so important for Physicists.

HallsofIvy

Homework Helper
I would think Euclid's proof would be sufficient.

matt grime

Homework Helper
Maxos said:
Very interesting but not so important for Physicists.
is it not? surely it is good to know things like this since ther are links between random matrices (spectral stuff) and the riemann zeta fucntion which is very heavily dependent on prime numbers.

Maxos

Yes you are right.

krab

Ahem! Back to the question of how to explain to NON-mathematicians. My experience is that it is useless to describe concepts with which we are already familiar; their eyes just glaze over. Simpler is to motivate it with the 7 bridges of Konigsberg. Then state that topology allows one to prove that the goal of crossing each bridge exactly once is impossible, so no one wastes time trying. Everyone understands about the value of not wasting time.

matt grime

Homework Helper
toplogy? i thought that was elementary graph theory: it contains 3 odd nodes or is it only 1?

Again, right.

fourier jr

matt grime said:
toplogy? i thought that was elementary graph theory: it contains 3 odd nodes or is it only 1?
it's a topological thing because the solution doesn't depend on the distances of the bridges from each other, nor the lengths of the bridges. the only thing to be concerned with is the connectivity properties. that's what the wikipedia thing said anyway.

matt grime

Homework Helper
Hmm, most odd. It is only a personal view but I've never considered graphs as anything other than combinatorics. Indeed I have never seen a graph theory theorem refer to any topology (perhaps the author is using it a nontechnical sense?) of the graph (ie not defining open subsets of the graph). Labels are largely unhelpful I admit, and one can certainly use topological ideas in graph theory as one can in many parts of mathematics.

Part of graph theory is concerned with "rigidity" and if a graph may be embedded in the plane.

I really can't agree with it, on reflection. I can't disagree with the assertion that graph theory is not bothered with how things are embedded in some space, merely the data of vertices and edges, but I don't see how that makes it topology. It seems to merely that it is merely a statement that we have realized these problems can be described in mathematical terms.

If you look at the graph theory (mathematics) link at the bottom of the page you'll find that the word topology doesn't appear at all in the description.

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