Summary: NOTE ON THE CONSTRUCTIBLE
SETS OF A TOPOLOGICAL SPACE
Jean­Paul Allouche
Introduction
Some twenty years ago, I tried to ``compute'' in a purely formal way as many
relations as possible involving the classical symbols of general topology, like A for
the closure of the set A,
ffi
A for the interior of A and all usual well­known other nota­
tions. In particular I obtained a characterization of the sets in the Boolean algebra
generated by the closed and the open sets, that I never published. Recently, R.
Mneimn'e, writing a book on Group Actions (see [9]), told me that the constructible
sets, (i.e., the sets in the Boolean algebra generated by the open and the closed
subsets of a topological space), are a useful tool in classical Algebraic Geometry:
for example Chevalley proved that the orbits of an algebraic affine group operat­
ing algebraically on an algebraic affine variety are locally closed (for the Zariski
topology) by proving first that the image of a polynomial map between algebraic
varieties is a constructible set, (see the paper of Cartan­Chevalley [3], or the book
of Borel [2], see also the book of Hartshorne [6, p. 94] or the book of Humphreys
[8, 4.4 p. 33]). For other occurrences of locally closed sets one can read the book of

Source: Allouche, Jean-Paul - Laboratoire de Recherche en Informatique, Université de Paris-Sud 11

Collections: Computer Technologies and Information Sciences