A kw-space X is a Hausdorff quotient of a locally compact, δ-compact Hausdorff space. A theorem of Morita's describes the structure of X when the quotient map is closed, but in 2010 a question of Arkhangel'skii's highlighted the lack of a corresponding theorem for non-closed quotient maps (even from subsets of Rn). Arkhangel'skii's specific question had in fact been answered by Siwiec in 1976, but a general structure theorem for kw-spaces is still lacking. We introduce pure quotient maps, extend Morita's theorem to these, and use Fell's topology to show that every quotient map can be 'purified' (and thus every kw-space is the image of a pure quotient map). This clarifies the structure of arbitrary kw-spaces and gives a fuller answer to Arkhangel'skii's question.