Extendability of Classes of Maps and New Properties of Upper Sets

We continue to study upper sets ${\widetilde{A}=\{(x,r)\in A\times R_+ :\exists y\in A\setminus\{x\}, r=|x-y|\}}We continue to study upper sets (A)\tilde]={(x,r) ? A×R₊ :$y ? A\{x}, r=|x-y|}{\widetilde{A}=\{(x,r)\in A\times R_+ :\exists y\in A\setminus\{x\}, r=|x-y|\}} equipped by hyperbolic metric. We define analogous of quasiconvexity, simply connectedness and nearlipschitz functions. We give a new definition of quasisymmetry as nearlipschitz characteristic on (A)\tilde]{\widetilde{A}}. In the final part in terms of upper sets we give the following extension property of A ì R²{A\subset R^2}. For 0 ￡ e ￡ d{0\le\varepsilon\le \delta}, each (1+e){(1+\varepsilon)}-bilipschitz map f : A → R ² has an extension to a (1+Ce){(1+C\varepsilon)}-bilipschitz map F : R ² → R ².