Let \(X_F\) denote the space of orderings of a formally real field \(F\). For every field extension \(F_1/F_0\), one can define a restriction map \(\varepsilon_{F_1/F_0}: X_{F_1}\to X_{F_0}\), assuming \(\varepsilon_{F_1/F_0}(L):= L\cap F_0\) for every positive cone \(L\) in \(F_1\). R. Elman, T. Y. Lam and A. R. Wadsworth [J. Reine Angew. Math. 306, 7–27 (1979; Zbl 0398.12019)] proved that \(\varepsilon_{F_1/F_0}\) is a continuous and open mapping, provided \(F_1/F_0\) is finitely generated.
Using the method of elimination of quantifiers the author proves that for every finitely generated field extension \(F_1/F_0\) the restriction map \(\varepsilon_{F_1/F_0}\) has a continuous section (i.e., there is a continuous mapping \(\sigma\colon Y\to X_{F_1}\) such that \(\varepsilon_{F_1/F_0}\circ \sigma = \operatorname{id}_Y\), where \(Y := \varepsilon_{F_1/F_0}(X_{F_1}))\).
This result is applied to solve some problems concerning the RRC-fields. Moreover, the general problem of existence of continuous sections of mappings of Boolean spaces is discussed.