Semialgebraic and semianalytic sets
Cahiers du séminaire d'histoire des mathématiques, Série 2, Tome 1 (1991), pp. 59-70. 
@article{CSHM_1991_2_1__59_0,
     author = {Ruiz, Jesus M.},
     title = {Semialgebraic and semianalytic sets},
     journal = {Cahiers du s\'eminaire d'histoire des math\'ematiques},
     pages = {59--70},
     publisher = {Institut Henri Poincar\'e, S\'eminaire d'histoire des math\'ematiques : Paris},
     volume = {Ser. 2, 1},
     year = {1991},
     mrnumber = {1122205},
     zbl = {0746.14023},
     language = {en},
     url = {http://archive.numdam.org/item/CSHM_1991_2_1__59_0/}
}
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Ruiz, Jesus M. Semialgebraic and semianalytic sets. Cahiers du séminaire d'histoire des mathématiques, Série 2, Tome 1 (1991), pp. 59-70. http://archive.numdam.org/item/CSHM_1991_2_1__59_0/

[L] Lojasiewicz S. : Ensembles semianalytiques, I.H.E.S. (prépublication) ( 1964). Much more recent is the book

[B C R] Bochnak J., Coste M., Roy M.-F.: Géométrie algébrique réelle. Berlin, Heidelberg, New York, Springer ( 1987). The first bounds on the number of inequalities needed to describe a semialgebraic set appeared in | MR | Zbl

[B1] Bröcker L.: Minimale Erzeugung von Positivbereich. Geom. Dedicate 16, 335-350 ( 1984). Finally the sharpest estimations were obtained in | MR | Zbl

[S] Scheiderer C. : Stability index of real varieties. Invent. Math. 97, 467-483 ( 1989). For more information on the semialgebraic case, an excellent reference is the recent survey | MR | Zbl

[B2] Bröcker L. : On basic semialgebraic sets. To apperar in Geom. Dedicata. Concerning the semianalytic case, Lojasiewicz's Lecture Notes quoted above is again the classical reference for local results. The global ones have appeared in the following papers

[R1] Ruiz J. : On Hilbert's 17th problem an real Nullstellensatz for global analytic functions. Math. Z. 190, 447-459 ( 1985). | MR | Zbl

[A B R 1] Andradas C., Bröcker L., Ruiz R. : Minimal generation of basic open semianalytic sets. Invent. Math. 92, 409-430 ( 1988), | MR | Zbl

[R 2] Ruiz J. : On the connected components of a global semianalytic set. Journal Reine Angew. Math. 392, 137-144 ( 1988). | MR | Zbl

[R 3] Ruiz J. : On the topology of global semianalytic sets. In Real analytic and algebraic geometry, Proc. Trento 1988, M. Galbiati, A. Tognoli (Eds.) Springer-Verlag LNM 1420, 237-246. Finally, there is an abstract theory of real constructible sets that generalizes both the algebraic and the analytic cases. A symmetric presentation of it is the subject of the forthcoming book. | MR | Zbl

[A B R 2] Andradas C., Bröcker L., Ruiz J., Real constructible sets.