Ernest Michael: How to Choose Without Choosing Badly
In 1956 a young topologist at the University of Washington published a theorem about making choices. Given a rule that assigns to each point not a single value but a whole set of acceptable values, when can you pick one from each — continuously, so that nearby points get nearby choices? The answer he gave is now used every time an economist proves an equilibrium exists or a control theorist proves a system can be steered. Fifty years later a journal devoted an entire special issue to what had grown from it, and called 1956 "the year of birth of this theory."
Zürich, Berlin, The Hague, New York
Michael was born in Zürich on 26 August 1925, the son of Jacob and Erna Michael, an Ashkenazi Jewish family.
The next fourteen years read as a map of a continent going wrong. The family moved from Berlin in 1932 — the year before Hitler took power, which is to say they read the situation early and correctly. They went to The Hague. In 1939, with the Netherlands months from invasion, they emigrated again, to New York. Michael would be described afterwards as American, of Swiss and German origin; each of those words was purchased with a departure.
He graduated from the Horace Mann School at fifteen. He went to Cornell, but the war interrupted him: he served aboard the USS Kwajalein from 1944 to 1946, returning to take his BA in 1947. A master's from Harvard followed in 1948, and in 1951 a PhD from the University of Chicago under Irving Segal, with a dissertation on locally multiplicatively-convex topological algebras.
Forty-One Years in Seattle
In 1952 Michael joined the mathematics department at the University of Washington. He was assistant professor from 1952 to 1956, associate professor from 1956 to 1960, and professor from 1960 until his retirement in 1993 — forty-one years in one department, in a life whose first fourteen had contained three countries.
He travelled from that base rather than moving again: four separate periods as a visitor at the Institute for Advanced Study, a year at ETH Zürich in 1973–74 — back to the city of his birth — and a year at the University of Stuttgart in 1978–79.
Twenty questions, eight minutes on the clock, and a percentile measured against everyone who has taken it. No sign-up.
Take the IQ test →The Selection Theorem
The result of 1956 concerns set-valued maps, sometimes called correspondences or multifunctions.
An ordinary function gives you one output per input. A set-valued map gives you a whole set of them: for each price, the set of quantities a firm might optimally produce; for each state of a machine, the set of controls that are permissible. A *selection* is an ordinary function that picks one element from each of those sets. Selections always exist if you accept the axiom of choice — but they will generally be wildly discontinuous, and a discontinuous choice is useless for the purposes people actually have.
Michael's selection theorem gives conditions under which a continuous selection exists: roughly, that the domain is paracompact and the map is lower semicontinuous with closed convex values. The theorem is the reason a great deal of modern analysis works. Fixed-point arguments in mathematical economics, the theory of differential inclusions, optimisation and control all rely at some point on being able to choose continuously from a moving set.
He did not stop at the theorem. He founded the whole theory of continuous selections around it, and the 1956 paper is properly regarded as that field's origin.
The Michael Line
His other famous construction is a counterexample, and it is one of the most quoted in general topology.
Topologists have long wanted properties to be well-behaved under products — if two spaces are nice, their product ought to be nice. The Michael line kills that hope for a specific and important pairing. It is a paracompact space whose product with the space of irrational numbers is not even normal.
Counterexamples of this quality do a particular job in mathematics. They mark the boundary of the possible, and they save everyone who comes afterwards from wasting years attempting a proof of something false. Michael's name also attaches to Michael's product topology and to results on closed subspaces, and he published over a hundred papers in general topology, in the *Transactions* and the *Annals* and *Duke*, on hyperspaces of subsets, paracompactness, continuous selections and the behaviour of normal spaces.
Why Ernest Is Called a Genius
The strongest evidence is durability of a specific and unusual kind. A single paper published in 1956 was still generating enough new mathematics fifty years later that *Topology and Its Applications* devoted a special issue to the anniversary. Very few results earn that.
The cognitive quality involved is worth naming precisely, because it is not raw computational power. Michael's gift was for finding the exact hypotheses — the minimal conditions under which a desirable thing is true, paired with the sharp examples showing what happens the moment you relax them. The selection theorem and the Michael line are the same talent facing in opposite directions: one maps where the ground is solid, the other marks precisely where it gives way. Mathematicians of this type are rarer than problem-solvers and more useful to a field's long-term health, because they define its geography.
The honest counter-case is that Michael's celebrated work is narrow in surface area. General topology in the second half of the twentieth century was not where the discipline's centre of gravity sat; his major honour was inclusion in the inaugural class of American Mathematical Society Fellows in 2012, a large and collective distinction that came fifty-six years after the theorem. He won no Wolf Prize, no Steele Prize, no national academy membership recorded in the accessible sources. No colleague is on record calling him a genius. Much of his selection theorem's fame is borrowed from downstream users — economists and control theorists who cite it as a lemma without engaging its topology at all. A fair verdict is that Michael was a mathematician of exceptional precision who found one theorem so exactly right that it outlived the fashion of his own field.
Legacy
Michael retired in 1993 and died on 29 April 2013, aged eighty-seven, in Seattle.
The theorem is in every graduate text on set-valued analysis. The Michael line is in every topology course that discusses products. Both are taught, most of the time, without anybody mentioning that their author was a Jewish boy who left Berlin in 1932, crossed an ocean ahead of an invasion, served in the Pacific, and then sat down in a room in Seattle for forty-one years and worked out precisely when it is possible to choose well.
Achievements
- Held posts at University of Washington
- Fields: mathematical analysis and topology
