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We fix once and for all a real closed base field R. By a variety X over R we mean a separated algebraic scheme X over R. For all problems attacked here we could equally well assume that X is also reduced, since we are only interested in the set X(R) of rational points of X. Notice that for every closed point x of X the residue class field K(x) = OJmx either coincides with R, i.e. x is rational, or is isomorphic to R(>/--T). We call the points x of X with K(X) = R the real points of X and the other closed points the complex points of X. R is a topological field, a basis of open sets being given by the open intervals

Semialgebraic topology over a real closed field I: Paths and components in the set of rational points of an algebraic variety