- #1

octol

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What do you all think about that?

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- Thread starter octol
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- #1

octol

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What do you all think about that?

- #2

Maxos

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

- #3

fourier jr

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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

- #4

HallsofIvy

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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).

- #5

matt grime

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

- #6

Galileo

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In the end they declared I was nuts.

- #7

matt grime

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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

- #10

arildno

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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.

- #11

PhilG

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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.

- #12

fourier jr

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other way around, the norm is automatically a metric if you write d(x,y) = ||x-y||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.

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- #13

PhilG

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

- #14

fourier jr

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HallsofIvy said:

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.

- #15

fourier jr

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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...)PhilG said:Right. I am asking if you can do calculus with JUST a metric, rather than a norm.

- #16

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Here are some other "What is Topology?" urls:

http://www.humboldt.edu/~mef2/Presentations/HSU Colloquia/colloq3_02/Outline.html

http://www.math.niu.edu/~rusin/known-math/index/54-XX.html [Broken]

http://www.mathsci.appstate.edu/classdescriptions/topology/topologyad.html [Broken]

http://www.math.wayne.edu/~rrb/topology.html

http://www.shef.ac.uk/~pm1nps/Wurble.html [Broken]

http://math.pepperdine.edu/kiga/topology.html

http://www.kolumbus.fi/justal/bits/math/topology.htm

http://www.humboldt.edu/~mef2/Presentations/HSU Colloquia/colloq3_02/Outline.html

http://www.math.niu.edu/~rusin/known-math/index/54-XX.html [Broken]

http://www.mathsci.appstate.edu/classdescriptions/topology/topologyad.html [Broken]

http://www.math.wayne.edu/~rrb/topology.html

http://www.shef.ac.uk/~pm1nps/Wurble.html [Broken]

http://math.pepperdine.edu/kiga/topology.html

http://www.kolumbus.fi/justal/bits/math/topology.htm

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- #17

PhilG

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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.

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

- #18

Haelfix

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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.

- #19

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 {a

Same thing with limits of functions. If f: M-> N is a function from topological space M to topological space N, x

That, of course, would give the same definition of "continuous function" as the one you give.

- #20

George Jones

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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

- #21

PhilG

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HallsofIvy said:If f: M-> N is a function from topological space M to topological space N, x_{0}is in M, then limit f(x)= b (as x->x_{0}), b in N, if and only if for every open set V containing b, there exist an open set U containing x_{0}such that f(U) is a subset of V.

f(x

- #22

matt grime

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- #23

PhilG

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So the answer to my question

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

- #24

EnumaElish

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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?PhilG said:Right. I am asking if you can do calculus with JUST a metric, rather than a norm.

- #25

EnumaElish

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Okay, how about derivative?matt grime said:

- #26

PhilG

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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:R

[tex]f'(a) = \lim_{h \rightarrow 0} \frac{f(a+h) - f(a)}{h}[/tex]

doesn't make sense because you can't divide a vector in R

[tex]\lim_{|h| \rightarrow 0} \frac{|f(a+h) - f(a) - \lambda(h)|}{|h|} = 0[/tex]

to define the derivative of f at a as the unique linear map λ : R

- #27

fourier jr

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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.

no, derivative is defined as a limit, at least in calculusEnumaElish said:Okay, how about derivative?

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.

- #28

matt grime

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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.

- #29

roger

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matt grime said:

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 ?

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roger

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- #31

matt grime

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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.

- #32

roger

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matt,

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

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

- #33

matt grime

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- #34

PhilG

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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?

- #35

Maxos

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

[tex]

f(\vec{x_0}+\vec{h})= \sum_{k=0}^n \frac {{d^k}f|_{\vec{x_0}}(\vec{h})} {k!} + o(|\vec{h}|^n)[/tex]

for [tex]\vec{h} \rightarrow \vec{0}

[/tex]

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

Saying: [tex]g(x)=o(f(x))[/tex] for [tex]x \rightarrow x_0[/tex] means that

[tex]\lim_{x\rightarrow x_0} \frac {g(x)} {f(x)} = 0[/tex]

[tex]

f(\vec{x_0}+\vec{h})= \sum_{k=0}^n \frac {{d^k}f|_{\vec{x_0}}(\vec{h})} {k!} + o(|\vec{h}|^n)[/tex]

for [tex]\vec{h} \rightarrow \vec{0}

[/tex]

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

Saying: [tex]g(x)=o(f(x))[/tex] for [tex]x \rightarrow x_0[/tex] means that

[tex]\lim_{x\rightarrow x_0} \frac {g(x)} {f(x)} = 0[/tex]

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