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On the topology of global semianalytic sets. (English) Zbl 0723.14041
Real analytic and algebraic geometry, Proc. Conf., Trento/Italy 1988, Lect. Notes Math. 1420, 237-246 (1990).
[For the entire collection see Zbl 0686.00007.]
Refining his arguments in several previous works, the author solves fundamental and interesting problems affirmatively on global semianalytic sets under rather general assumptions.
A subset Z of a real analytic manifold M is called a global semianalytic set if it is a finite union of subsets of the form \(\{z\in M;f_ 1(z)>0,...,f_ s(z)>0\), \(g(z)=0\}, f_ i,g\in {\mathcal O}(M)\), where \({\mathcal O}(M)\) is the ring of analytic functions on M. - For instance, in this paper, the following is proved: Let Z be a global semianalytic set. If cl(Z)-Z (respectively Z-int(Z)) is relatively compact, then cl(Z) (respectively int(Z)) is also a global semianalytic set.
The theory of real spectrum of a ring is basic for the proofs: it is precisely applied to the ring \(A={\mathcal O}(M)\), the strict localization \(A_{\alpha}\) of which by any prime cone \(\alpha\) is an excellent ring.
14P15 Real-analytic and semi-analytic sets
32B20 Semi-analytic sets, subanalytic sets, and generalizations
14F45 Topological properties in algebraic geometry
Zbl 0686.00007