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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.

Inhaltsverzeichnis

Frontmatter

Chapter I. Preliminaries

Abstract
In this section, we give definitions of a Whitney stratification, a tube system, a vector field on a stratification, isomorphisms between Whitney stratifications, etc., and show their properties needed later, particularly I.1.13, with complete proofs. Some of our definitions are a little different from the usual ones, e.g. [G-al]. We modify the definitions to suit them to our purpose. We treat special topics unknown even to singularity specialists. We use them in Chapter III. For this, we need the method of integration of vector fields, which may contradict our philosophy. This is because the theorems in Chapter III are stated in a more general situation than X. The X-versions of the results of this section and Chapter III, except 1.1.6 and 1.1.7, can be proved without the method of integration. Note that the X-versions work in the C r category, r a positive integer (see Chapter II).
Masahiro Shiota

Chapter II. X-Sets

Abstract
In this chapter, r always denotes a positive integer, and smoothness means Cr smoothness. (However, the theorems II and II′ hold for r = ∞ and ω if we assume the property (II.1.8) for such r, which is clear by their proofs.) For its axiomatic treatment, the following definition of a subanalytic set, which is equivalent to the definition in §I.2, is adequate. The family of all subanalytic sets in Euclidean spaces is the smallest family & of subsets of Euclidean spaces which satisfies the following axioms
Masahiro Shiota

Chapter III. Hauptvermutung for Polyhedra

Abstract
By (II. 1.15) and (II. 1.19), an X-homeomorphism between compact X-subsets of R n is a strong C r isomorphism for some C r Whitney xstra-tifications. This property seems to be the key of the X-Hauptvermutung (uniqueness of x-triangulations of xsets). Hence we conjecture.
Masahiro Shiota

Chapter IV. Triangulations of X-Maps

Abstract
Let XR n and YR m be locally closed X-sets and let f : XY be an X-map. A C0 X-triangulation of f is a quadruplet of X-polyhedra X0R n ’ and Y0R m ’ and X-homeomorphisms π : X0X and τ : Y0Y such that τ-1 o f o π is PL. We call a C0 X-triangulation (X0, X0, π, τ) a C0 R-X-triangulation if Y is a polyhedron, Y0 = Y and τ = id. In connection with the preceeding chapters, it may be natural to treat an (R-)X-triangulation of f which we define assuming, in addition, that π and τ are of class C r , r > 0, on each simplex of some simplicial decompositions of X0 and Y0 respectively. However, I cannot prove the results of this chapter in the terms of an (R-)X-triangulation, and I think that a C0 (R-) X-triangulation is more natural than an (R-) Xtriangulation in itself. We define naturally a C0 X-stratification of an X-set in a Euclidean space. Here note that the stratification is finite locally at each point of the Euclidean space and each stratum is not only an X-set and a C0 manifold but also a C0 X-submanifold of the Euclidean space (i.e., locally X-homeomorphic to a Euclidean space).
Masahiro Shiota

Chapter V. Y-Sets

Abstract
The simplest family of subsets of Euclidean spaces which we can treat systematically in topological position may be the family of semilinear sets. We generalize the concept of X so that this family is an example. Let D be a family of subsets of Euclidean spaces which satisfies the following four axioms.
Masahiro Shiota

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