Semialgebraic space
Encyclopedia
In mathematics
, especially in real algebraic geometry
, a semialgebraic space is a space which is locally isomorphic to a semialgebraic set
.
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.
A semialgebraic space is a locally ringed space
which is locally isomorphic to Rn with its sheaf of semialgebraic functions.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, especially in real algebraic geometry
Real algebraic geometry
In mathematics, real algebraic geometry is the study of real algebraic sets, i.e. real-number solutions to algebraic equations with real-number coefficients, and mappings between them ....
, a semialgebraic space is a space which is locally isomorphic to a semialgebraic set
Semialgebraic set
In mathematics, a semialgebraic set is a subset S of Rn for some real closed field R defined by a finite sequence of polynomial equations and inequalities , or any finite union of such sets. A semialgebraic function is a function with semialgebraic graph...
.
Definition
Let U be an open subset of Rn for some n. A semialgebraic function on U is defined to be a continuousContinuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
-valued function on U whose restriction to any semialgebraic set
Semialgebraic set
In mathematics, a semialgebraic set is a subset S of Rn for some real closed field R defined by a finite sequence of polynomial equations and inequalities , or any finite union of such sets. A semialgebraic function is a function with semialgebraic graph...
contained in U has a graph
Graph of a function
In mathematics, the graph of a function f is the collection of all ordered pairs . In particular, if x is a real number, graph means the graphical representation of this collection, in the form of a curve on a Cartesian plane, together with Cartesian axes, etc. Graphing on a Cartesian plane is...
which is a semialgebraic subset of the product space Rn×R. This endows Rn with a sheaf
of semialgebraic functions.A semialgebraic space is a locally ringed space
which is locally isomorphic to Rn with its sheaf of semialgebraic functions.See also
- Semialgebraic setSemialgebraic setIn mathematics, a semialgebraic set is a subset S of Rn for some real closed field R defined by a finite sequence of polynomial equations and inequalities , or any finite union of such sets. A semialgebraic function is a function with semialgebraic graph...
- Real algebraic geometryReal algebraic geometryIn mathematics, real algebraic geometry is the study of real algebraic sets, i.e. real-number solutions to algebraic equations with real-number coefficients, and mappings between them ....
- Real closed ringReal closed ringIn mathematics, a real closed ring is a commutative ring A thatis a subring of a product of real closed fields, which is closed undercontinuous semi-algebraic functions defined over the integers.- Examples of real closed rings :...