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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'''<sup>''n''</sup> for some ''n''. A '''semialgebraic function''' on ''U'' is defined to be a [[continuous function|continuous]] [[real number|real]]-valued function on ''U'' whose restriction to any [[semialgebraic set]] contained in ''U'' has a [[graph of a function|graph]] which is a semialgebraic subset of the product space '''R'''<sup>''n''</sup>×'''R'''. This endows '''R'''<sup>''n''</sup> with a sheaf <math>\mathcal{O}_{\mathbf{R}^n}</math> 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]] <math>(X, \mathcal{O}_X)</math> which is locally isomorphic to '''R'''<sup>''n''</sup> with its sheaf of semialgebraic functions.

==See also==
*[[Semialgebraic set]]
*[[Real algebraic geometry]]
*[[Real closed ring]]

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[[Category:Real algebraic geometry]]