On Tightness-Type Properties of the Space of Weakly Additive Functionals

We study tightness-type properties such as tightness, minitightness, and local density of the space of weakly additive functionals with finite support. We also investigate some generalizations of continuous functions. Furthermore, we present an extension of the functor of weakly additive functionals with finite support to the class of strictly $\tau $-continuous mappings. We introduce two extensions of the categories $\mathrm {Comp}$ and $\mathrm {Tych}$ (of compact and Tychonoff spaces, respectively). One of the main results of the paper is that the functor $O_n$ of weakly additive functionals with finite support preserves the tightness character of infinite compact spaces. In addition, we show that the local densities of the spaces $X$ and $O_n(X)$ coincide for any infinite compact space $X$.