Our effort to weaken algebraic assumptions led us to obtain
characterizations of C(X) as Riesz spaces, real l-groups, semi-affine
lattices and real lattices by using different techniques. We present a unified
approach valid for any “convenient” category. By setting equivalent
conditions to equi-uniform continuity, we obtain a characterization of
the lattice U(Y ) in parallel with that of C(X).
