Entangled points

The terms Limit point, Cluster point, Accumulation point don’t really match the concept point set topology is trying to capture.

As usual, the motivation for any topological concept (including this one) lies in the real numbers.

1 is a limit point of the open interval (0, 1) of real numbers. Any open interval containing 1 also contains elements of (0, 1). 1 is entangled with the set (0, 1) given the usual topology of the real line.

What is the usual topology of the real line? (E.g. how are its open sets defined) It’s the set of open intervals) and their infinite unions and their finite intersection.

In this topology no open set can separate 1 from the set ( 0, 1) — e.g. they are entangled.

So call 1 an entangled point.This way of thinking allows you to think of open sets as separators of points from sets.

Hausdorff thought this way, when he described the separation axioms (TrennungsAxioms) describing points and sets that open sets could and could not separate.

The most useful collection of open sets satisfy Trennungsaxiom #2 — giving a Hausdorff topological space. There are enough of them so that every two distinct points are contained in two distinct disjoint open sets.

Thinking of limit points as entangled points gives you a more coherent way to think of continuous functions between topological spaces. They never separate a set and any of its entangled points in the domain when they map them to the target space. At least to me, this is far more satisfactory (and actually equivalent) to continuity than the usual definition; the inverse of an open set in the target space is an open set in the domain.

Clarity of thought and ease of implementation are two very different things. It is much easier to prove/disprove that a function is continuous using the usual definition than using the preservation of entangled points.

Organic chemistry could certainly use some better nomenclature. Why not call an SN1 reaction (Substitution Nucleophilic 1) SN-pancake — as the 4 carbons left after the bond is broken form a plane. Even better SN2 should be called SN-umbrella, as it is exactly like an umbrella turning inside out in the wind.

The strangeness of mathematical proof

I’ver written about Urysohn’s Lemma before and a copy of that post will be found at the end. I decided to plow through the proof since coming up with it is regarded by Munkres (the author of a widely used book on topology) as very creative. Here’s how he introduces it

“Now we come to the first deep theorem of the book,. a theorem that is commonly called the “Urysohn lemma”. . . . It is the crucial tool used in proving a number of important theorems. . . . Why do we call the Urysohn lemma a ‘deep’ theorem? Because its proof involves a really original idea, which the previous proofs did not. Perhaps we can explain what we mean this way: By and large, one would expect that if one went through this book and deleted all the proofs we have given up to now and then handed the book to a bright student who had not studied topology, that student ought to be able to go through the book and work out the proofs independently. (It would take a good deal of time and effort, of course, and one would not expect the student to handle the trickier examples.) But the Uyrsohn lemma is on a different level. It would take considerably more originality than most of us possess to prove this lemma.”
”

I’m not going to present the proof just comment on one of the tools used to prove it. This is a list of all the rational numbers found in the interval from 0 to 1, with no repeats.

Munkres gives the list at its start and you can see why it would list all the rational numbers. Here it is

0, 1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5 . . .

Note that 2/4 is missing (because 2 divides into 4 leaving a whole number). It would be fairly easy to write a program to produce the list, but a computer running the program would never stop. In addition it would be slow, because to avoid repeats given a denominator n, it would include 1/n and n-1/n in the list, but to rule out repeats it would have to perform n-2 divisions. It it had a way of knowing if a number was prime it could just put in 1/prime, 2/prime , , , (prime -1)/n without the division. But although there are lists of primes for small integers, there is no general way to find them, so brute force is required. So for 10^n, that means 10^n – 2 divisions. Once the numbers get truly large, there isn’t enough matter in the universe to represent them, nor is there enough time since the big bang to do the calculations.

However, the proof proceeds blithely on after showing the list — this is where the strangeness comes in. It basically uses the complete list of rational numbers as indexes for the infinite number of open sets to be found in a normal topological space. The proof below refers to the assumption of infinite divisibility of space (inherent in the theorem on normal topological spaces), something totally impossible physically.

So we’re in the never to be seen land of completed infinities (of time, space, numbers of operations). It’s remarkable that this stuff applies to the world we inhibit, but it does, and anyone wishing to understand physics at a deep level must come to grips with mathematics at this level.

Here’s the old post

Urysohn’s Lemma

The above quote is from one of the standard topology texts for undergraduates (or perhaps the standard text) by James R. Munkres of MIT. It appears on page 207 of 514 pages of text. Lee’s text book on Topological Manifolds gets to it on p. 112 (of 405). For why I’m reading Lee see https://luysii.wordpress.com/2012/09/11/why-math-is-hard-for-me-and-organic-chemistry-is-easy/.

Well it is a great theorem, and the proof is ingenious, and understanding it gives you a sense of triumph that you actually did it, and a sense of awe about Urysohn, a Russian mathematician who died at 26. Understanding Urysohn is an esthetic experience, like a Dvorak trio or a clever organic synthesis [ Nature vol. 489 pp. 278 – 281 ’12 ].

Clearly, you have to have a fair amount of topology under your belt before you can even tackle it, but I’m not even going to state or prove the theorem. It does bring up some general philosophical points about math and its relation to reality (e.g. the physical world we live in and what we currently know about it).

I’ve talked about the large number of extremely precise definitions to be found in math (particularly topology). Actually what topology is about, is space, and what it means for objects to be near each other in space. Well, physics does that too, but it uses numbers — topology tries to get beyond numbers, and although precise, the 202 definitions I’ve written down as I’ve gone through Lee to this point don’t mention them for the most part.

Essentially topology reasons about our concept of space qualitatively, rather than quantitatively. In this, it resembles philosophy which uses a similar sort of qualitative reasoning to get at what are basically rather nebulous concepts — knowledge, truth, reality. As a neurologist, I can tell you that half the cranial nerves, and probably half our brains are involved with vision, so we automatically have a concept of space (and a very sophisticated one at that). Topologists are mental Lilliputians trying to tack down the giant Gulliver which is our conception of space with definitions, theorems, lemmas etc. etc.

Well one form of space anyway. Urysohn talks about normal spaces. Just think of a closed set as a Russian Doll with a bright shiny surface. Remove the surface, and you have a rather beat up Russian doll — this is an open set. When you open a Russian doll, there’s another one inside (smaller but still a Russian doll). What a normal space permits you to do (by its very definition), is insert a complete Russian doll of intermediate size, between any two Dolls.

This all sounds quite innocent until you realize that between any two Russian dolls an infinite number of concentric Russian dolls can be inserted. Where did they get a weird idea like this? From the number system of course. Between any two distinct rational numbers p/q and r/s where p, q, r and s are whole numbers, you can always insert a new one halfway between. This is where the infinite regress comes from.

For mathematics (and particularly for calculus) even this isn’t enough. The square root of two isn’t a rational number (one of the great Euclid proofs), but you can get as close to it as you wish using rational numbers. So there are an infinite number of non-rational numbers between any two rational numbers. In fact that’s how non-rational numbers (aka real numbers) are defined — essentially by fiat, that any series of real numbers bounded above has a greatest number (think 1, 1.4, 1.41, 1.414, defining the square root of 2).

What does this skullduggery have to do with space? It says essentially that space is infinitely divisible, and that you can always slice and dice it as finely as you wish. This is the calculus of Newton and the relativity of Einstein. It clearly is right, or we wouldn’t have GPS systems (which actually require a relativistic correction).

But it’s clearly wrong as any chemist knows. Matter isn’t infinitely divisible, Just go down 10 orders of magnitude from the visible and you get the hydrogen atom, which can’t be split into smaller and smaller hydrogen atoms (although it can be split).

It’s also clearly wrong as far as quantum mechanics goes — while space might not be quantized, there is no reasonable way to keep chopping it up once you get down to the elementary particle level. You can’t know where they are and where they are going exactly at the same time.

This is exactly one of the great unsolved problems of physics — bringing relativity, with it’s infinitely divisible space together with quantum mechanics, where the very meaning of space becomes somewhat blurry (if you can’t know exactly where anything is).

Interesting isn’t it?

A book recommendation, not a review

My first encounter with a topology textbook was not a happy one. I was in grad school knowing I’d leave in a few months to start med school and with plenty of time on my hands and enough money to do what I wanted. I’d always liked math and had taken calculus, including advanced and differential equations in college. Grad school and quantum mechanics meant more differential equations, series solutions of same, matrices, eigenvectors and eigenvalues, etc. etc. I liked the stuff. So I’d heard topology was cool — Mobius strips, Klein bottles, wormholes (from John Wheeler) etc. etc.

So I opened a topology book to find on page 1

A topology is a set with certain selected subsets called open sets satisfying two conditions
l. The union of any number of open sets is an open set
2. The intersection of a finite number of open sets is an open set

Say what?

In an effort to help, on page two the book provided another definition

A topology is a set with certain selected subsets called closed sets satisfying two conditions
l. The union of a finite number number of closed sets is a closed set
2. The intersection of any number of closed sets is a closed set

Ghastly. No motivation. No idea where the definitions came from or how they could be applied.

Which brings me to ‘An Introduction to Algebraic Topology” by Andrew H. Wallace. I recommend it highly, even though algebraic topology is just a branch of topology and fairly specialized at that.

Why?

Because in a wonderful, leisurely and discursive fashion, he starts out with the intuitive concept of nearness, applying it to to classic analytic geometry of the plane. He then moves on to continuous functions from one plane to another explaining why they must preserve nearness. Then he abstracts what nearness must mean in terms of the classic pythagorean distance function. Topological spaces are first defined in terms of nearness and neighborhoods, and only after 18 pages does he define open sets in terms of neighborhoods. It’s a wonderful exposition, explaining why open sets must have the properties they have. He doesn’t even get to algebraic topology until p. 62, explaining point set topological notions such as connectedness, compactness, homeomorphisms etc. etc. along the way.

This is a recommendation not a review because, I’ve not read the whole thing. But it’s a great explanation for why the definitions in topology must be the way they are.

It won’t set you back much — I paid. $12.95 for the Dover edition (not sure when).

Urysohn’s Lemma

“Now we come to the first deep theorem of the book,. a theorem that is commonly called the “Urysohn lemma”.  . . .  It is the crucial tool used in proving a number of important theorems. . . .  Why do we call the Urysohn lemma a ‘deep’ theorem?  Because its proof involves a really original idea, which the previous proofs did not.  Perhaps we can explain what we mean this way:  By and large, one would expect that if one went through this book and deleted all the proofs we have given up to now and then handed the book to a bright student who had not studied topology, that student ought to be able to go through the book and work out the proofs independently.  (It would take a good deal of time and effort, of course, and one would not expect the student to handle the trickier examples.)  But the Uyrsohn lemma is on a different level.  It would take considerably more originality than most of us possess to prove this lemma.”

The above quote is  from  one of the standard topology texts for undergraduates (or perhaps the standard text) by James R. Munkres of MIT. It appears on  page 207 of 514 pages of text.  Lee’s text book on Topological Manifolds gets to it on p. 112 (of 405).  For why I’m reading Lee see https://luysii.wordpress.com/2012/09/11/why-math-is-hard-for-me-and-organic-chemistry-is-easy/.

Well it is a great theorem, and the proof is ingenious, and understanding it gives you a sense of triumph that you actually did it, and a sense of awe about Urysohn, a Russian mathematician who died at 26.   Understanding Urysohn is an esthetic experience, like a Dvorak trio or a clever organic synthesis [ Nature vol. 489 pp. 278 – 281 ’12 ].

Clearly, you have to have a fair amount of topology under your belt before you can even tackle it, but I’m not even going to state or prove the theorem.  It does bring up some general philosophical points about math and its relation to reality (e.g. the physical world we live in and what we currently know about it).

I’ve talked about the large number of extremely precise definitions to be found in math (particularly topology).  Actually what topology is about, is space, and what it means for objects to be near each other in space.  Well, physics does that too, but it uses numbers — topology tries to get beyond numbers, and although precise, the 202 definitions I’ve written down as I’ve gone through Lee to this point don’t mention them for the most part.

Essentially topology reasons about our concept of space qualitatively, rather than quantitatively.  In this, it resembles philosophy which uses a similar sort of qualitative reasoning to get at what are basically rather nebulous concepts — knowledge, truth, reality.   As a neurologist, I can tell you that half the cranial nerves, and probably half our brains are involved with vision, so we automatically have a concept of space (and a very sophisticated one at that).  Topologists are mental Lilliputians trying to tack down the giant Gulliver which is our conception of space with definitions, theorems, lemmas etc. etc.

Well one form of space anyway.  Urysohn talks about normal spaces.  Just think of a closed set as a Russian Doll with a bright shiny surface.  Remove the surface, and you have a rather beat up Russian doll — this is an open set.  When you open a Russian doll, there’s another one inside (smaller but still a Russian doll).  What a normal space permits you to do (by its very definition), is insert a complete Russian doll of intermediate size, between any two Dolls.

This all sounds quite innocent until you realize that between any two Russian dolls an infinite number of concentric Russian dolls can be inserted.  Where did they get a weird idea like this?  From the number system of course.  Between any two distinct rational numbers p/q and r/s where p, q, r and s are whole numbers, you can  always insert a new one halfway between.  This is where the infinite regress comes from.

For mathematics (and particularly for calculus) even this isn’t enough.  The square root of two isn’t a rational number (one of the great Euclid proofs), but you can get as close to it as you wish using rational numbers.  So there are an infinite number of non-rational numbers between any two rational numbers.  In fact that’s how non-rational numbers (aka real numbers) are defined — essentially by fiat, that any series of real numbers bounded above has a greatest number (think 1, 1.4, 1.41, 1.414, defining the square root of 2).

What does this skullduggery have to do with space?  It says essentially that space is infinitely divisible, and that you can always slice and dice it as finely as you wish.  This is the calculus of Newton and the relativity of Einstein.  It clearly is right, or we wouldn’t have GPS systems (which actually require a relativistic correction).

But it’s clearly wrong as any chemist knows. Matter isn’t infinitely divisible, Just go down 10 orders of magnitude from the visible and you get the hydrogen atom, which can’t be split into smaller and smaller hydrogen atoms (although it can be split).

It’s also clearly wrong as far as quantum mechanics goes — while space might not be quantized, there is no reasonable way to keep chopping it up once you get down to the elementary particle level.  You can’t know where they are and where they are going exactly at the same time.

This is exactly one of the great unsolved problems of physics — bringing relativity, with it’s infinitely divisible space together with quantum mechanics, where the very meaning of space becomes somewhat blurry (if you can’t know exactly where anything is).

Interesting isn’t it?