Semialgebraic space

In mathematics, especially in real algebraic geometry, a semialgebraic space is a space which is locally isomorphic to a semialgebraic set.

Definition

Let "U" be an open subset of R^"n" for some "n". A semialgebraic function on "U" is defined to be a continuous real-valued function on "U" whose restriction to any semialgebraic set contained in "U" has a graph which is a semialgebraic subset of the product space R^"n"×R. This endows R^"n" with a sheaf $mathcal\{O\}\_\{mathbf\{R\}^n\}$ of semialgebraic functions.

A semialgebraic space is a locally ringed space $(X,\; mathcal\{O\}\_X)$ which is locally isomorphic to R^"n" with its sheaf of semialgebraic functions.

ee also

*Semialgebraic set
*Real algebraic geometry