Quotient spaces for semialgebraic equivalence relations (Q1080904)

Let \(X\subset R^ n\) be an affine semi-algebraic set over a real closed field R, and let \(E\subset X\times X\) be a closed, semi-algebraic equivalence relation. This paper proves that the space of equivalence classes is also an affine semi-algebraic set if the projection \(E\to X\) is proper. For example, this hypothesis holds if \(X\) is closed and bounded in \(R^ n\) or if \(E\) is the orbit equivalence relation associated to a continuous semi-algebraic action of a closed, bounded group on \(X\). When \(E\times X\) is not proper, the paper gives some necessary conditions for the existence of an affine quotient space.