Real-compact spaces and the real line orthogonal (Q2630437)

In category theory a morphism \(f:A\to B\) and an object \(X\) are called orthogonal if the map from \(\mathrm{hom}(B, X)\) to \(\mathrm {hom}(A, X)\) that sends \(g\) to \(g\circ f\) is bijective. For a class of morphisms \(\Sigma\) (respectively objects) the class \(\Sigma^{\perp}\) contains all objects (respectively morphisms) that are orthogonal to all morphisms (respectively objects) in \(\Sigma\). The authors prove several properties of continuous maps between topological spaces that are orthogonal to the real line. Moreover, they prove that \(\mathbf{Hewitt}^{\perp}=\mathbb{R}^{\perp}\) and \(\mathbf{Hewitt}=\mathbb{R}^{\perp \perp}\), where \(\mathbf{Hewitt}\) is the class of all realcompact topological spaces.