In his thesis work, published as a journal article in 1931, Freudenthal introduced the concept of an
end of a
topological space. Ends are intended to capture the intuitive idea of a direction in which the space extends to infinity, but have a precise mathematical formulation in terms of covers of the space by nested sequences of
compact sets. Ends remain of great importance in
topological group theory, Freudenthal's motivating application, and also in other areas of mathematics such as the study of
minimal surfaces. In 1936, while working with Brouwer, Freudenthal proved the
Freudenthal spectral theorem on the existence of uniform approximations by
simple functions in
Riesz spaces. In 1937 he proved the
Freudenthal suspension theorem, showing that the
suspension operation on topological spaces shifts by one their low-dimensional
homotopy groups; this result was important in understanding the
homotopy groups of spheres (since every sphere can be formed topologically as a suspension of a lower-dimensional sphere) and eventually formed the basis of
stable homotopy theory. The
Freudenthal magic square is a construction in
Lie algebra developed by Freudenthal (and independently by
Jacques Tits) in the 1950s and 1960s, associating each Lie algebra to a pair of
division algebras. In 1968, Freudenthal founded the journal,
Educational Studies in Mathematics (ESM). Becoming one of the top-rated journals in the field of mathematics education, ESM was focused on publishing research around finding better ways to teach mathematics. Later in his life, Freudenthal focused on elementary
mathematics education. In the 1970s, his single-handed intervention prevented the Netherlands from following the worldwide trend of "
new math". He interpreted mathematics as a human activity where students should open a scientific eye on the world around them, mathematizing real situations, in a context that makes sense for the students. This approach is called
Realistic Mathematics Education (RME). Freudenthal published the
Impossible Puzzle, a mathematical puzzle that appears to lack sufficient information for a solution, in 1969. He also designed a
constructed language,
Lincos, to make possible
communication with extraterrestrial intelligence. ==Selected publications==