In 2010 a question of Arhangel'skii's highlighted a gap in the knowledge of k_��-spaces. His specific question had in fact been answered by Siwiec in 1976, but the highlighted gap still remains. We introduce the simple idea of pure quotient maps, extend Morita's theorem to these, and use Fell's topology to show that every quotient map onto a k_��-space can be 'purified'; and thus fill the gap, elucidate the structure of k_��-spaces, and obtain a fuller answer to Arhangel'skii's question.

The paper was submitted under another title. The readers should instead refer to arXiv:2108.05202