Let $Y$ be a locally compact, $\sigma$-compact, first countable Hausdorff space and $q:Y\to X$ a quotient map with $X$ Hausdorff. Suppose that $X$ is locally compact. Is $X$ first countable at a dense set of points?
It was shown here that global first countability can fail: Is a locally compact Hausdorff quotient of a locally compact $\sigma$-compact first countable Hausdorff space always first countable?
Here is a partial answer. Suppose that $Y$ is hereditarily Lindelof (so that the assumption of first countability is implied and can be omitted). Then this is equivalent to every open subset of $Y$ being Lindelof, and this is equivalent, in the presence of local compactness, to every open subset of $Y$ being $\sigma$-compact. But this property passes to the quotient space $X$, which is thus also hereditarily Lindelof, and assumed to be locally compact, and must thus be first countable.
