THE EQUIVALENCE CONDITIONS FOR SEMIALGEBRAICALLY PROPER MAPS Park, Dae Heui;
In this paper we compare the notion of proper map in the category of topological spaces with that in the category of semialgebraic sets. To do this, we find some equivalence conditions for semialgebraically proper maps. In particular, we prove that a continuous semialgebraic map is semialgebraically proper if and only if it is proper. Moreover, we compare the semialgebraically proper map with the proper map in the sense of Delfs and Knebush .
semialgebraic set;semialgebraic map;proper map;
J. Bochnak, M. Coste and M.-F. Roy, Real Agebraic Geometry, Erg. der Math. und ihrer Grenzg., vol. 36, Springer-Verlag, Berlin Heidelberg, 1998.
N. Bourbaki, General Toplogy, Assison-Wesley, 1966.
G. W. Brumfiel, Quotient space for semialgebraic equivalence relation, Math. Z. 195 (1987), 69-78.
H. Delfs and M. Knebusch, Semialgebraic topology over a real closed field II: Basic theory of semialgebraic spaces, Math. Z. 178 (1981), 175-213.
J. Dugundji, Toplogy, Allyn and Bacon, Boston, 1966.
R. Robson, Embedding semi-algebraic spaces, Math. Z. 183 (1983), 365-370.
C. Scheiderer, Quotients of semi-algebraic spaces, Math. Z. 201 (1989), 249-271.