  In 2010 a question of Arhangel'skii's highlighted a gap in the knowledge of
k_{\omega}-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_{\omega}-space can be 'purified'; and
thus fill the gap, elucidate the structure of k_{\omega}-spaces, and obtain a
fuller answer to Arhangel'skii's question.
