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.

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).