Weakly Semialgebraic Spaces
The book is the second part of an intended three-volume treatise on semialgebraic topology over an arbitrary real closed field R. In the first volume (LNM 1173) the category LSA(R) or regular paracompact locally semialgebraic spaces over R was studied. Th
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1367
Manfred Knebusch
Weakly Semialgebraic Spaces
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Lecture Notes in Mathematics Edited by A. Oold and B. Eckmann
1367
Manfred Knebusch
Weakly Semialgebraic Spaces
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Author
Manfred Knebusch Fakultat fur Mathematik, Universitat Regensburg 8400 Regensburg, Federal Republic of Germany
Mathematics Subject Classification (1980): 14G30, 54E99, 54E60, 55005, 55N 10, 55N20, 55P05, 55P 10 ISBN 3-540-50815-5 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-50815-5 Springer-Verlag New York Berlin Heidelberg
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© Springer-Verlag Berlin Heidelberg 1989 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210
Introduction This is the second in a chain of (hopefully) three volumes devoted to an explication of the fundamentals of semialgebraic topology over an arbitrary real closed field R. We refer the uninitiated reader to the preface of the first volume [LSA)1 and some other papers cited there to get an idea of the program we have in mind with the term "semialgebraic topology" as a basis of real algebraic geometry.
Let us roughly recall what has been achieved in the first volume and where we stand now. As we explained in [LSA), the "good" locally semialgebraic spaces, which fortunately seem to suffice for most applications, are the regular paracompact ones. These are precisely those locally semialgebraic spaces which can be triangulated (I.4.8 and 11.4.4)2. Moreover, any locally finite family of locally semialgebraic sets in such a space can be triangulated simultaneously (II.4.4). This fact seems to be the key result for many proofs in [LSA).
We accomplished less work on the triangulation of locally semialgebraic maps. Here our main result has been the triangulability of finite maps (II.6.13). Much more can probably be done, as is to be expected by the book [V] of Verona, but we do not pursue this line of investigation in the present volume. {Verona works over
m
and uses transcendental
techniques.}
cf. the references
2
This refers to Example 4.8 in Chapter I and Theorem 4.4 in Chapter II of [LSA]. The main body of this volume starts with Chapter IV. The signs I, II, III refer to the chapters of [LSA].
IV
On the other hand we obtained in Chapter II of [LSA] a fairly detailed picture of the various possibilities how to "complete" a regular paracompact space M, i.e. to embed M densely
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