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# 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.

(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 \( (X, \mathcal{O}_X) \) which is locally isomorphic to Rn with its sheaf of semialgebraic functions.

See also

Semialgebraic set

Real algebraic geometry

Real closed ring

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