### A dense subsemigroup of S(R) generated by two elements

We develop a calculus for the oscillation index of Baire one functions using gauges analogous to the modulus of continuity.

Normal spaces are characterized in terms of an insertion type theorem, which implies the Katětov-Tong theorem. The proof actually provides a simple necessary and sufficient condition for the insertion of an ordered pair of lower and upper semicontinuous functions between two comparable real-valued functions. As a consequence of the latter, we obtain a characterization of completely normal spaces by real-valued functions.

We determine the size levels for any function on the hyperspace of an arc as follows. Assume Z is a continuum and consider the following three conditions: 1) Z is a planar AR; 2) cut points of Z have component number two; 3) any true cyclic element of Z contains at most two cut points of Z. Then any size level for an arc satisfies 1)-3) and conversely, if Z satisfies 1)-3), then Z is a diameter level for some arc.

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...

We study relations between the cellularity and index of narrowness in topological groups and their ${G}_{\delta}$-modifications. We show, in particular, that the inequalities $in\left({\left(H\right)}_{\tau}\right)\le {2}^{\tau \xb7in\left(H\right)}$ and $c\left({\left(H\right)}_{\tau}\right)\le {2}^{{2}^{\tau \xb7in\left(H\right)}}$ hold for every topological group $H$ and every cardinal $\tau \ge \omega $, where ${\left(H\right)}_{\tau}$ denotes the underlying group $H$ endowed with the ${G}_{\tau}$-modification of the original topology of $H$ and $in\left(H\right)$ is the index of narrowness of the group $H$. Also, we find some bounds for the complexity of continuous real-valued functions $f$ on an arbitrary $\omega $-narrow group $G$ understood...