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This paper deals with questions of how many compact subsets of certain kinds it takes to cover the space ${}^{\omega }\omega$ of irrationals, or certain of its subspaces. In particular, given $f\in {}^{\omega }(\omega \setminus \left\{0\right\})$, we consider compact sets of the form ${\prod }_{i\in \omega }{B}_i$, where $|B_i|=f\left(i\right)$ for all, or for infinitely many, $i$. We also consider “$n$-splitting” compact sets, i.e., compact sets $K$ such that for any $f\in K$ and $i\in \omega$, $\left|\right\{g\left(i\right):g\in K,g\upharpoonright i=f\upharpoonright i\left\}\right|=n$.

A space $X$ is functionally countable (FC) if for every continuous $f:X\to ℝ$, $\left|f\right(X\left)\right|\le \omega$. 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 $f:X\to ℝ$, $\left|f\right(Y\left)\right|\le \omega$; $X$ is 3-FS if for every continuous $f:X\to ℝ$, there is a dense subspace $Y\subset X$ such that $\left|f\right(Y\left)\right|\le \omega$. We give examples distinguishing...

A space is monotonically Lindelöf (mL) if one can assign to every open cover $𝒰$ a countable open refinement $r\left(𝒰\right)$ so that $r\left(𝒰\right)$ refines $r\left(𝒱\right)$ whenever $𝒰$ refines $𝒱$. We show that some countable spaces are not mL, and that, assuming CH, there are countable mL spaces that are not second countable.

If $X$ is a space, then the $we\left(X\right)$ of $X$ is the cardinal $min\{\alpha :$ If $𝒰$ is an open cover of $X$, then there exists $A\subseteq X$ such that $\left|A\right|=\alpha$ and $St(A,𝒰)=X\}$. In this note, we show that if $X$ is a normal space such that $\left|X\right|=𝔠$ and $we\left(X\right)=\omega$, then $X$ does not have a closed discrete subset of cardinality $𝔠$. We show that this result cannot be strengthened in ZFC to get that the extent of $X$ is smaller than $𝔠$, even if the condition that $we\left(X\right)=\omega$ is replaced by the stronger condition that $X$ is separable.

We characterize spaces with $\sigma$-$n$-linked bases as specially embedded subspaces of separable spaces, and derive some corollaries, such as the $𝐜$-productivity of the property of having a $\sigma$-linked base.