Functor of $\mathrm{OS}_{\sigma}$-semiadditive, $\sigma$-smooth functionals

In this paper, we introduce the functor $\mathrm{OS}_{\sigma}$ of semiadditive $\sigma$-smooth functionals into the category of Tikhonov spaces $\mathrm{Tych}$, which extends the functor $\mathrm{OS}:\mathrm{Comp}\to\mathrm{Comp}$ of semiadditive functionals. We prove that the functor $\mathrm{OS}_{\sigma}:\mathrm{Tych}\to\mathrm{Tych}$ maps $Z$-embeddings into embeddings and the space $\mathrm{OS}_{\sigma}(X)$ is closed in the space of weakly additive $\sigma$-smooth functionals; in particular, $\mathrm{OS}_{\sigma}(X) $ is Hewitt-complete for any Tikhonov space $X\in\mathrm{Tych}$.