Real analytic sets in Euclidean space (Le. sets defined locally at each point of Euclidean
space by the vanishing of an analytic function) were first investigated in the 1950's by H.
Cartan [Car] H. Whitney [WI-3] F. Bruhat [W-B] and others. Their approach was to derive
information about real analytic sets from properties of their complexifications. After some
basic geometrical and topological facts were established however the study of real analytic
sets stagnated. This contrasted the rapid develop ment of complex analytic geometry which
followed the groundbreaking work of the early 1950's. Certain pathologies in the real case
contributed to this failure to progress. For example the closure of -or the connected
components of-a constructible set (Le. a locally finite union of differ ences of real
analytic sets) need not be constructible (e. g. R - {O} and 3 2 2 { (x y z) E R : x = zy2
x + y2 -=I- O} respectively). Responding to this in the 1960's R. Thorn [Thl] S. Lojasiewicz
[LI 2] and others undertook the study of a larger class of sets the semianalytic sets which
are the sets defined locally at each point of Euclidean space by a finite number of ana lytic
function equalities and inequalities. They established that semianalytic sets admit Whitney
stratifications and triangulations and using these tools they clarified the local topological
structure of these sets. For example they showed that the closure and the connected components
of a semianalytic set are semianalytic.
