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...

Kechris and Louveau in [5] classified the bounded Baire-1 functions, which are defined on a compact metric space $K$, to the subclasses ${ℬ}_1^{\xi }\left(K\right)$, $\xi <{\omega }_1$. In [8], for every ordinal $\xi <{\omega }_1$ we define a new type of convergence for sequences of real-valued functions ($\xi$-uniformly pointwise) which is between uniform and pointwise convergence. In this paper using this type of convergence we obtain a classification of pointwise convergent sequences of continuous real-valued functions defined on a compact metric space $K$, and...

Let $(L,𝒯)$ be a Tychonoff (regular) paratopological group or algebra over a field or ring $K$ or a topological semigroup. If $nw(L,𝒯)\le \tau$ and $nw\left(K\right)\le \tau$, then there exists a Tychonoff (regular) topology ${𝒯}^{*}\subseteq 𝒯$ such that $w(L,{𝒯}^{*})\le \tau$ and $(L,{𝒯}^{*})$ is a paratopological group, algebra over $K$ or a topological semigroup respectively.