Insertion of a function belonging to a certain subclass of \(\mathbb R^ X\). (Q1413663)

Let \(X\) be a topological space and \(E(X,{\mathcal R})\) the uniformly closed linear subspace of \({\mathcal R}^X\) containing all constant functions such that \(\min(f,g),\max(f,g) \in E(X,{\mathcal R})\) for each constant \(g\) and \(f \in E(X,{\mathcal R})\). Let \(P_1,P_2 \supset E(X,{\mathcal R})\) be classes of functions such that for each \(f \in P_i\), \(i=1,2\), and \(g\in E(X,{\mathcal R})\) the sum \(f + g \in P_i\). A space \(X\) has the weak \(E\)-insertion property (the \(E\)-insertion property, resp., the strong \(E\)-insertion property, resp.) for \((P_1,P_2)\) iff for any \(g\in P_1\) and \(f\in P_2\) with \(g \leq f\) (\(g < f\), \(g \leq f\)) there is a function \(h \in E(X,{\mathcal R})\) such that \(g\leq h \leq f\) (\(g < h < f\), \(g\leq h\leq f\) and if \(g(x) < f(x)\) then \(g(x) < h(x) < f(x)\)). The author compares these properties and describes some cases where these properties are equivalent.