Semialgebraic space

Let U be an open subset of Rⁿ 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 Rⁿ×R. This endows Rⁿ with a sheaf ${\displaystyle {\mathcal {O}}_{\mathbf {R} ^{n}}}$ of semialgebraic functions.

(For example, any polynomial mapping between semialgebraic sets is a semialgebraic function, as is the maximum of two semialgebraic functions.)

A semialgebraic space is a locally ringed space ${\displaystyle (X,{\mathcal {O}}_{X})}$ which is locally isomorphic to Rⁿ with its sheaf of semialgebraic functions.^[1]

• Real closed ring