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.

Sn2 — It’s a gas

Sn2 reactions are a lot more complicated than as taught in orgo 101 (at least in the gas phase). The classic mechanism is very easy to teach to students, it’s just an umbrella turning inside out in the wind. A current article in Science (vol. 352 pp. 32 – 33 1 April ’16) shows how complicated things can be when the reaction is carried out in the gas phase. Mechanisms illustrated include rebound stripping, frontside attack, ion-dipole complex, roundabout, hydrogen bond complex, frontside complex and double inversion.

Why study Sn2 in the gas phase? One reason is to sharpen computational and theoretical methods to be able to predict reaction rates (in gas phase reactions). I was surprised on looking up Rice-Ramsperger-Kassel-Marcus theory to find out how old it was. Back in the 60’s it was taught to us without any names attached. One assumes that before and after reaction the ion molecule complexes are trapped in potential wells. It is assumed that vibrational energies in the complex are quickly distributed to ‘equilibrium’ in the complexes so that detailed computation of rates can be carried out.

Is this of any use to the chemist actually reacting molecules in solution? Other than by sharpening computational tools, I don’t see how it can be given the present state of the art.

Gas phase kineticists are starting to try, but they’ve got a very long way to go. “Stepwise addition of solvent molecules to the bare reactant anion offers a bottom up approach to learn more about the transition of chemical reactions from the gas to liquid phase. To investigate the role of solvation in Sn2 reactions Otto et al. have performed crossed molecular beam studies of the microsolvated” Sn2 reaction (e.g. the approaching anion solvated with all of one or two waters). “The results show that “the dynamics differ dramatically from the unsolved anion.”