Tag Archives: continuous function

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.

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.


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).

Why math is hard (for me) and organic chemistry is easy

I’ve been reading a lot of hard core math lately (I’ll explain why at the end), along with Clayden et al’s new edition of their fabulous Organic Chemistry text.  The level of sophistication takes a quantum jump about 2/3 of the way through (around pp. 796) and is probably near to the graduate level.  The exercise is great fun, but math and orgo require quite distinct ways of thinking.  Intermixing both on a daily basis brought home just how very different they are.

First off, the concepts in organic chemistry are fuzzy.  On p. 796 the graph of the Karplus relationship between J splitting in NMR and the dihedral angle of the hydrogens being split is shown.  It’s a continuous curve as the splitting is maximal at 180, zero at 90 and somewhat less than maximal at 0 degrees.

There is nothing like this in math.  Terms are defined exactly and the logic is that of true, false and the excluded middle (e.g. things are either true or false).   Remember the way that the square root of 2 was proved not to be the ratio of two whole numbers.  It was assumed that it could be done, and than it was shown no matter how you sliced it, a contradiction was reached.   The contradiction then implied that the opposite was true — if the negative of a proposition leads to a contradiction (it’s false) than the proposition must be true.  Math is full of proofs like this.Or if you are trying to prove A implies B, proving the contrapositive ( not B implies not A) will do just as well.  You never see stuff like this in orgo.

There just aren’t that many concepts in organic chemistry, even though the details of each and every reaction are beyond the strongest memory.  The crucial points are to have the orbitals of the various atoms firmly in mind and where they are in space.  This tells you how molecules will or won’t react, or how certain conformations will be stable (see anomeric effect).  Entropy in physics is a very subtle concept, but pretty obvious as used by organic chemists.  Two molecules are better than one etc. etc.  Also you see these concepts over and over.  Everything you study (just about) has carbon in it.  Chair and boat, cis and trans, exo and endo become part of you, without thinking much about them.

Contrast this with math.  I’m currently reading “Introduction to Topological Manifolds” (2nd. Edition) by John M. Lee.  I’ve got about 34 pages of notes on the first 95 pages (25% of the text), and made a list of the definitions I thought worth writing down — there are 170 of them.  Each is quite precise.  A topological embedding is (1) a continuous function (2) a surjective function (3) a homeomorphism.  No more no less.  Remove any one of the 3 (examples are given) and you no longer have an embedding.  The definitions are abstract for the most part, and far from intuitive.  That’s because  the correct definitions were far from obvious even to the mathematicians formulating them.  Hubbard’s excellent book on Vector Calculus says that it took some 200 years for the correct definition of continuity to be thrashed out.  People were arguing about what a function actually was 100 years ago.

As you read you are expected to remember exactly (or look up) the  170 or so defined concepts and use them in a proof.  So when you read a bit of Lee’s book, I’m always stopping and asking myself  ‘did I really understand what I just read’?  Clayden isn’t at all like that — Oh that’s just an intramolecular Sn2, helped because of the Thorpe Ingold effect, which is so obvious it shouldn’t be given a name.

Contrast this with:

After defining topological space, open set, closed set, compact, Hausdorff, continuous, closed map, you are asked to show that a continuous map from a compact topological space to a Hausdorff topological space is a closed map, and that such a map, if surjective as well is an embedding.   To get even close to the proof you must be able to hold all this in your head at once.  You should also remember that you proved that in a Hausdorff space compact sets are closed.

No matter how complicated the organic problem, you can always look at the molecule, and use the fabulous spatial processing capacity your brain gives you.   The interpretation of NMR spectra in terms of conformation certainly requires logical thinking — it’s sort of organic Sudoku.

I imagine a mathematician would have problems with the fuzzy concepts of organic chemistry.  Anslyn and Dougherty take great pains to show you why some reactions fall between Sn1 and Sn2, or E1cb.

So why am I doing this?  Of course there’s the why climb Everest explanation — because it’s there, big and hard, and maybe you can’t do it.  That’s part of it, but not all.  For reasons unknown, I’ve always like math, even though not terribly good at it.  Then there’s the surge for the ego should I be able to go through it all proving that I don’t have Alzheimer’s at 74.5 (at least not yet).  Then there is the solace (yes solace) that math provides.  Topology is far from new to me in 2011.  I started reading Hocking and Young back in ’70 when I was a neurology resident, seeing terrible disease, being unable to help most of those I saw, and ruminating about the unfairness of it all.  Thinking about math took me miles away (and still does), at least temporarily.  When I get that far away look, my wife asks me if I’m thinking about math again.  She’s particularly adamant about not doing this when I’m driving with her (or by myself).

The final reason, is that I went to college with a genius.  I met him at our 50th reunion after reading his bio in our 50th reunion book.  I knew several self-proclaimed geniuses back then, and a lot a physics majors, but he wasn’t one of them.  At any rate, he’s still pumping out papers on relativity with Stephen Hawking, and his entries in the index of the recent biography by Kitty Ferguson take up almost as many entries as Hawking himself.  He’s a very nice guy and agreed to answer questions from time to time.  But to understand the physics you need to really understand the math, and not just mouth it.

In particular, to understand gravity, a la relativity, you have to know how mass bends 4 dimensional space-time. This means you must understand curvature on manifolds, which means you must understand smooth manifolds, which means that you must understand topological manifolds which is why I’m reading Lee’s book.
So perhaps when the smoke clears, I might have something intelligent to say to my classmate.