We define a class of spaces which is more extensive than the class of BCO spaces and which counts among its members some that are not even first countable, and show that this more extensive class of spaces nevertheless intersects the class of paracompact Hausdorff spaces at precisely the class of metrizable spaces as does the class of BCO spaces, thus extending a theorem of Arhangel'skil. We further show that this extension of Arhangel'skil's result has gone the farthest in the sense that any

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... ass of spaces that meets the paracompact spaces at precisely the metrizable spaces must, among the Hausdorff spaces, be smaller than the class we have defined.