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Let $Y$ and $Z$ be two arbitrary fixed topological spaces, $C(Y,Z)$ the set of all continuous maps from $Y$ to $Z$, and ${𝒪}_Z(Y)$ the set consisting of all open subsets $V$ of $Y$ such that $V=f^{-1}(U)$, where $f\in C(Y,Z)$ and $U$ is an open subset of $Z$. In this paper we continue the study of the $𝒜$-proper and $𝒜$-admissible topologies on ${𝒪}_Z(Y)$, where $𝒜$ is an arbitrary family of spaces, initiated in [6] and we offer new results concerning the finest $X$-proper topology $\tau (\{X\})$ on ${𝒪}_Z(Y)$ for several metrizable spaces $X$.

The universality problem focuses on finding universal spaces in classes of topological spaces. Moreover, in “Universal spaces and mappings” by S. D. Iliadis (2005), an important method of constructing such universal elements in classes of spaces is introduced and explained in details. Simultaneously, in “A topological dimension greater than or equal to the classical covering dimension” by D. N. Georgiou, A. C. Megaritis and F. Sereti (2017), new topological dimension is introduced and studied, which...