Semialgebraic space

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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 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 O R n {\displaystyle {\mathcal {O}}_{\mathbf {R} ^{n}}} {\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 ( X , O X ) {\displaystyle (X,{\mathcal {O}}_{X})} {\displaystyle (X,{\mathcal {O}}_{X})} which is locally isomorphic to Rn with its sheaf of semialgebraic functions.[1]

See also

References

  1. Delfs, Hans; Knebusch, Manfred (1981). "Semialgebraic Topology over a Real Closed Field II: Basic Theory of Semialgebraic Spaces" (PDF). Mathematische Zeitschrift. 178 (2): 175–213. Retrieved 17 March 2026.