 
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
