On [r, s]-Superporosity

The main goal of the paper is to characterize families of [r,s]-(upper) superporous subsets of R, which generalize well-known notions of superporosity and strong superporosity of subsets of R. Definitions and properties of [r,s]-superporosity are symmetric to definitions and properties of superporosity and strong superporosity. The purpose in all cases is to define small subsets of the line using the notion of porosity. Superporous sets preserve positive porosity, and strongly superporous sets preserve strong porosity; i.e., if E is superporous (correspondingly, E is strongly superporous), then for every x∈E and for every F such that porosity of F at x is greater than 0 (correspondingly, is equal to 1), porosity of E∪F at x is greater than 0 (correspondingly, is equal to 1). Taking arbitrary positive porosity, instead of 0 or 1, we obtain the symmetric definition as follows: [r,s]-superporosity for 0<r≤s<1 transfers s-porosity to r-porosity; i.e., if E is [r,s]-superporous, then for every x∈E and for every F such that porosity of F at x is not less than s, porosity of E∪F at x is not less than r. Even though the definition and properties of [r,s]-superporosity, superporosity and strong superporosity are symmetric and all of them consist of very small sets, the families of these sets are essentially different. In the paper, we focus on relationships between [r,s]-superporous sets for different indices [r,s]. Furthermore, we compare [r,s]-superporosity to superporosity and strong superporosity. We apply the notion of [r,s]-superporosity to find multipliers and adders of porouscontinuous functions.