Functor of weakly additive τ-smooth functionals and mappings

We present a criterion for the τ-smoothness of weakly additive order-preserving functionals and establish that the functor of τ-smooth weakly additive functionals preserves the perfectness of mappings. We prove that if a continuous mapping between Tikhonov spaces is such that the mapping between the corresponding spaces of weakly additive functionals generated by it is open, then the indicated mapping is also open. It is shown that the converse statement is true under certain additional conditions.