We examine when a space $X$ has a zero set universal parametrised by a metrisable space of minimal weight and show that this depends on the $\sigma$-weight of $X$ when $X$ is perfectly normal. We also show that if $Y$ parametrises a zero set universal for $X$ then $hL\left(X^n\right)\le hd\left(Y\right)$ for all $n\in \mathbb{N}$. We construct zero set universals that have nice properties (such as separability or ccc) in the case where the space has a $K$-coarser topology. Examples are given including an $S$ space with zero set universal parametrised by an $L$ space (and...

A subset $A$ of a metric space $(X,d)$ is central iff for every Katětov map $f:X\to ℝ$ upper bounded by the diameter of $X$ and any finite subset $B$ of $X$ there is $x\in X$ such that $f\left(a\right)=d(x,a)$ for each $a\in A\cup B$. Central subsets of the Urysohn universal space $𝕌$ (see introduction) are studied. It is proved that a metric space $X$ is isometrically embeddable into $𝕌$ as a central set iff $X$ has the collinearity property. The Katětov maps of the real line are characterized.