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What do you all think about that?

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- #1

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What do you all think about that?

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

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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).

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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).

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Galileo

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In the end they declared I was nuts.

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matt grime

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Icebreaker

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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

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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.

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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.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.

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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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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||

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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.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).

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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.

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

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 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.

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HallsofIvy

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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.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.

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.

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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

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f(xHallsofIvy 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.

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matt grime

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- #23

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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?

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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.

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EnumaElish

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Okay, how about derivative?matt grime said:

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