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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 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×R. This endows R 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 R with its sheaf of semialgebraic functions.

See also

• Semialgebraic set
• Real algebraic geometry
• Real closed ring

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