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Geometry of Subanalytic and Semialgebraic Sets (Paperback, Softcover reprint of the original 1st ed. 1997)
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Geometry of Subanalytic and Semialgebraic Sets (Paperback, Softcover reprint of the original 1st ed. 1997)
Series: Progress in Mathematics, 150
Expected to ship within 10 - 15 working days
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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.
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