### ${\aleph}_{0}$-spaces and images of separable metric spaces.

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A space $X$ is functionally countable if $f\left(X\right)$ is countable for any continuous function $f:X\to \mathbb{R}$. We will call a space $X$ exponentially separable if for any countable family $\mathcal{F}$ of closed subsets of $X$, there exists a countable set $A\subset X$ such that $A\cap \bigcap \mathcal{G}\ne \varnothing $ whenever $\mathcal{G}\subset \mathcal{F}$ and $\bigcap \mathcal{G}\ne \varnothing $. Every exponentially separable space is functionally countable; we will show that for some nice classes of spaces exponential separability coincides with functional countability. We will also establish that the class of exponentially separable spaces has...

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 $f:X\to \mathbb{R}$ 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 $\mathbb{U}$ (see introduction) are studied. It is proved that a metric space $X$ is isometrically embeddable into $\mathbb{U}$ as a central set iff $X$ has the collinearity property. The Katětov maps of the real line are characterized.

We show that exponential separability is an inverse invariant of closed maps with countably compact exponentially separable fibers. This implies that it is preserved by products with a scattered compact factor and in the products of sequential countably compact spaces. We also provide an example of a $\sigma $-compact crowded space in which all countable subspaces are scattered. If $X$ is a Lindelöf space and every $Y\subset X$ with $\left|Y\right|\le {2}^{{\omega}_{1}}$ is scattered, then $X$ is functionally countable; if every $Y\subset X$ with $\left|Y\right|\le {2}^{\U0001d520}$ is scattered, then...

A space $X$ is functionally countable (FC) if for every continuous $f:X\to \mathbb{R}$, $\left|f\right(X\left)\right|\le \omega $. The class of FC spaces includes ordinals, some trees, compact scattered spaces, Lindelöf P-spaces, $\sigma $-products in ${2}^{\kappa}$, and some L-spaces. We consider the following three versions of functional separability: $X$ is 1-FS if it has a dense FC subspace; $X$ is 2-FS if there is a dense subspace $Y\subset X$ such that for every continuous $f:X\to \mathbb{R}$, $\left|f\right(Y\left)\right|\le \omega $; $X$ is 3-FS if for every continuous $f:X\to \mathbb{R}$, there is a dense subspace $Y\subset X$ such that $\left|f\right(Y\left)\right|\le \omega $. We give examples distinguishing...