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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]]
 
{{geometry-stub}}
 
[[Category:Real algebraic geometry]]

Revision as of 15:04, 8 January 2014

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 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 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 which is locally isomorphic to Rn with its sheaf of semialgebraic functions.

See also



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