On the connected components of a global semianalytic subset of an analytic surface.

*(English)*Zbl 1113.14037It is a well-known fact that a connected component and the closure of a semialgebraic set are again semialgebraic sets and that a similar statement holds true for semianalytic sets.

The author considers the problem of whether a union of connected components of a global semianalytic set is again global semianalytic. The answer to this problem was known to be positive for boundary bounded global semianalytic sets, see the book of C. Andradas, L. BrĂ¶cker and J. M. Ruiz [Constructible sets in real geometry. Ergebn. Math. Grenzgeb. 3, 33 (1996; Zbl 0873.14044)], and for global semianalytic subsets of a connected paracompact real manifold of dimension two, see [A. Castilla and C. Andradas, J. Reine Angew. Math. 475, 137–148 (1996; Zbl 0862.32003)].

In this paper it is proved that a union of connected components of a global semianalytic subset \(X\) of a coherent analytic subset of dimension two with affine normalization is again a global semianalytic subset. It must be noticed that a different proof of the same statement has been given by C. Andradas and the reviewer [Ill. J. Math. 48, No. 2, 519–537 (2004; Zbl 1077.14087)].

The author also proves partial results for connected components of a global semianalytic subset of a three-dimensional analytic manifold.

In order to get the main results the author proves some useful technical lemmas to deal with semianalytic subsets and also an interesting generalization of Thom’s lemma for convergent power series.

The author considers the problem of whether a union of connected components of a global semianalytic set is again global semianalytic. The answer to this problem was known to be positive for boundary bounded global semianalytic sets, see the book of C. Andradas, L. BrĂ¶cker and J. M. Ruiz [Constructible sets in real geometry. Ergebn. Math. Grenzgeb. 3, 33 (1996; Zbl 0873.14044)], and for global semianalytic subsets of a connected paracompact real manifold of dimension two, see [A. Castilla and C. Andradas, J. Reine Angew. Math. 475, 137–148 (1996; Zbl 0862.32003)].

In this paper it is proved that a union of connected components of a global semianalytic subset \(X\) of a coherent analytic subset of dimension two with affine normalization is again a global semianalytic subset. It must be noticed that a different proof of the same statement has been given by C. Andradas and the reviewer [Ill. J. Math. 48, No. 2, 519–537 (2004; Zbl 1077.14087)].

The author also proves partial results for connected components of a global semianalytic subset of a three-dimensional analytic manifold.

In order to get the main results the author proves some useful technical lemmas to deal with semianalytic subsets and also an interesting generalization of Thom’s lemma for convergent power series.

Reviewer: Antonio Diaz-Cano (Madrid)