Let $mathfrak{M}$ be a class of metric spaces. A metric space $Y$ is minimal $mathfrak{M}$-universal if every $Xinmathfrak{M}$ can be isometrically embedded in $Y$ but there are no proper subsets of $Y$ satisfying this property. We find conditions un
der which, for given metric space $X$, there is a class $mathfrak{M}$ of metric spaces such that $X$ is minimal $mathfrak{M}$-universal. We generalize the notion of minimal $mathfrak{M}$-universal metric space to notion of minimal $mathfrak{M}$-universal class of metric spaces and prove the uniqueness, up to an isomorphism, for these classes. The necessary and sufficient conditions under which the disjoint union of the metric spaces belonging to a class $mathfrak{M}$ is minimal $mathfrak{M}$-universal are found. Examples of minimal universal metric spaces are constructed for the classes of the three-point metric spaces and $n$-dimensional normed spaces. Moreover minimal universal metric spaces are found for some subclasses of the class of metric spaces $X$ which possesses the following property. Among every three distinct points of $X$ there is one point lying between the other two points.
For subsets of $mathbb{R}^+$ we consider the local right upper porosity and the local right lower porosity as elements of a cluster set of all porosity numbers. The use of a scaling function $mu:mathbb{N} to mathbb{R}^+$ provides an extension of the
concept of porosity numbers on subsets of $mathbb{N}$. The main results describe interconnections between porosity numbers of a set, features of the scaling functions and the geometry of so-called pretangent spaces to this set.
