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 $in\left({\left(H\right)}_{\tau }\right)\le 2^{\tau ·in\left(H\right)}$ and $c\left({\left(H\right)}_{\tau }\right)\le 2^{2^{\tau ·in\left(H\right)}}$ 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...

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...

We conduct an investigation of the relationships which exist between various generalizations of complete regularity in the setting of merotopic spaces, with particular attention to filter spaces such as Cauchy spaces and convergence spaces. Our primary contribution consists in the presentation of several counterexamples establishing the divergence of various such generalizations of complete regularity. We give examples of: (1) a contigual zero space which is not weakly regular and is not a Cauchy...

This note establishes that the familiar internal characterizations of the Tychonoff spaces whose rings of continuous real-valued functions are complete, or $\sigma$-complete, as lattice ordered rings already hold in the larger setting of pointfree topology. In addition, we prove the corresponding results for rings of integer-valued functions.

This paper deals with the existence of non constant real valued functions on a topological space X. The main results are related to closed covers and order properties.

A topological space is non-separably connected if it is connected but all of its connected separable subspaces are singletons. We show that each connected sequential topological space X is the image of a non-separably connected complete metric space X under a monotone quotient map. The metric $d_X$ of the space X is economical in the sense that for each infinite subspace A ⊂ X the cardinality of the set $d_X(a,b):a,b\in A$ does not exceed the density of A, $|d_X\left(A\times A\right)|\le dens\left(A\right)$. The construction of the space X determines a functor : Top...