First-countable spaces every quotient of which is orderable

We prove the following Main Theorem: Every Hausdorff quotient image of a first-countable Hausdorff topological space X is a linearly ordered topological space (LOTS) if and only if X is a metrizable space which is the union of a discrete subspace and a compact countable subspace. As a corollary we characterize 1) σ-compact spaces, 2) locally compact spaces, 3) separable spaces every quotient image of which is a LOTS. We examine also several natural examples of non-first-countable Hausdorff topological spaces X such that every quotient image of X is a LOTS.