We study the class of first-countable Lindel\"of scattered spaces, or "FLS" spaces. While every $T_3$ FLS space is homeomorphic to a subspace $\mathbb Q$, $T_2$ spaces turns out be surprisingly rich. Our investigation these reveals close ties $Q$-sets, Lusin sets, and their relatives, cardinals $\mathfrak{b}$ $\mathfrak{d}$. Many natural questions about turn independent $\mathsf{ZFC}$. prove th...

A space $Y$ is called an {em extension} of a space $X$, if $Y$ contains $X$ as a dense subspace. Two extensions of $X$ are said to be {em equivalent}, if there is a homeomorphism between them which fixes $X$ point-wise. For two (equivalence classes of) extensions $Y$ and $Y'$ of $X$ let $Yleq Y'$, if there is a continuous function of $Y'$ into $Y$ which fixes $X$ point-wise. An extension $Y$ ...

The minimum weight of a nowhere first-countable compact space of countable π-weight is shown to be κB, the least cardinal κ for which the real line R can be covered by κ many nowhere dense sets.