Urysohn lemma

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aguacate
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Urysohn lemma

Postby aguacate » Sun Dec 30, 2007 10:02 pm UTC

In my topology book by Munkres, the Urysohn lemma is described as "the first deep theorem of the book" because its proof involves a very original idea that very very few people could come up with on their own. I am a little self conscious on my basic proof skills and I feel like it is somehow significant to at least recognise the tricky ideas in proofs, and I do not in this one. I understand it as:

1. Identify the rational numbers with concentric sets surrounding the given closed set A and disjoint from the closed set B. Using induction, identify the simple ordering of the rational numbers with inclusion of such sets.

2. Use this identification to define a function from the normal space X to [0,1].

3. Show that the function is continuous, and satisfies f(A)=0, f(B)=1.


Which part or parts would you say makes for the very original idea?
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Re: Urysohn lemma

Postby skeptical scientist » Sun Dec 30, 2007 10:09 pm UTC

Well, the whole point of the proof is how you define the function f, so the idea you need is really how to define it. Once you have the definition, showing it works is not too bad. But the idea of defining a function by taking concentric subsets and identifying them with the rational numbers is not obvious.
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aguacate
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Re: Urysohn lemma

Postby aguacate » Sun Dec 30, 2007 10:44 pm UTC

Maybe I just didn't get the obviousness of the rest of the proofs in my topology course, so this one didn't seem like it was on a totally different level. That course kicked my ass. Although my teacher kept telling us that the the proofs were all trivial and the definitions made the class hard. Maybe I should go back and restudy what we learned.

Man, I don't even know why this is so interesting to me, but does anyone know a little history on what motivated this lemma in the first place? Did it just pop up out of the blue? (I can see how that would be quite an original feat).
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Re: Urysohn lemma

Postby skeptical scientist » Sun Dec 30, 2007 11:20 pm UTC

Sorry. I never took any point-set topology and learned mostly through other classes, so I can't really help you with the history or motivation. Part of the motivation probably just came from trying to work with real-valued continuous functions, since they let you deduce things from the topology on the reals, which is simple and easily understood and visualized.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.

"With math, all things are possible." —Rebecca Watson

Kalak_z
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Re: Urysohn lemma

Postby Kalak_z » Mon Dec 31, 2007 8:06 am UTC

Your professor is absolutely right. Most of the theorems are practically tautologies. The reason you, along with many other students, have trouble with this is because the conclusions of the theorems came first. Most of point set topology is an outgrowth of "I have this beautiful result from analysis, but do I need to assume that I am working in Rn?" And, from this they pulled out the important properties that they needed and gave them names.

On a more practical note, you will find that when you become more specialized, the topological results that you need to know will be stated again. If you work in Algebraic geometry or commutative algebra, you will talk about the Zariski topology, which is not T2, and what topological results apply. If you work in analysis, you will talk about weak and weak* topologies on dual spaces, which are not metrizable, and how compactness results can be used.

So, to answer your question, the ingenuity in topology was not in the results, but in the definitions. And, the weird counterexamples.

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Re: Urysohn lemma

Postby acidhak » Sun Jan 27, 2008 5:08 am UTC

Even though the original post has been pretty much wrapped up here, I must say... The way that he defined the function to make the proof work was maybe the first case in the book/ever where it wasn't about a straight-forward usual proof. If you tried to prove his lemma without knowing his function, it would most likely not get done. It was an ingenious step and the idea of using such a function was subsequently copied for use in hundreds/thousands of other theorems from topology to functional analysis. This kind of function, starting with him, is now considered a standard trick in a mathematicians bag of proof "trickery". Just like adding and subtracting something, then using Cauchy-Schwartz... obvious kind of trick once someone points it out.


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