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Algebra and topology, together
Henriksen spent his career on a single, very fertile question: how much of a space X can you read off from the ring C(X) of continuous real-valued functions on it? The answers, worked out with Leonard Gillman, John Isbell, Meyer Jerison and many others, became a field.
The work
Henriksen took his doctorate at the University of Wisconsin in 1951, supervised by Richard Bruck and William Eberlein, with a thesis on the ideal structure of the ring of entire functions and other function rings. The question of what the ideals of a function ring know about the underlying object stayed with him for the rest of his life.
With Leonard Gillman in the 1950s he introduced two classes of spaces that are still named in every text on the subject: the P-spaces, in which every prime ideal of C(X) is maximal, and the F-spaces, in which every finitely generated ideal is principal. The point was not the definitions but the method — a topological property and an algebraic one turning out to be the same thing seen from two sides. That work fed directly into Gillman and Jerison's Rings of Continuous Functions (1960), the book that fixed the subject's language.
With Isbell he studied compactifications; with Jerison, the space of minimal prime ideals of a commutative ring, a construction that reaches well beyond function rings. Later work with Jan Vermeer and Grant Woods on quasi-F covers, and with Jorge Martínez and Woods on prime z-ideals, kept the same two-sided habit going into the 2000s.
For pointfree topology this is the classical half of a conversation. Banaschewski's long project on C(L) for a frame L asks Henriksen's questions with the points removed — and much of what is known about the pointfree function ring is measured against what Henriksen and his collaborators proved about C(X).
A space is known by the functions it carries. Ask the right question of the ring, and the topology answers.
A life
Born.
Ph.D., University of Wisconsin, under R. H. Bruck and W. F. Eberlein.
At Purdue, where the work with Leonard Gillman on rings of continuous functions begins.
Joins Harvey Mudd College, Claremont, as Professor of Mathematics.
Retires from Harvey Mudd — and, like Banaschewski, keeps working in the field.
Dies in the autumn, aged 82.
A memorial conference on his life and work is held at Harvey Mudd on 27 March.
Selected works
Henriksen's bibliography is long and heavily collaborative. These are the papers most often cited as the foundations of the area.
With L. Gillman. Transactions of the AMS. Introduces P-spaces and sets out the ideal theory of C(X) that later work builds on.
With L. Gillman. Transactions of the AMS. The paper that defines F-spaces by an algebraic property of their function rings.
With J. R. Isbell. Duke Mathematical Journal. Compactifications studied through the functions they extend.
With M. Jerison. Transactions of the AMS. A topology on minimal primes that has since become standard well outside function rings.
Pointfree topology keeps asking what survives when the points are gone. For rings of functions, the thing it is checking against is very often a theorem of Henriksen's.