A note on countably paracompact spaces and metrization

As defined by Dowker [1], a topological space (X, 3) is called countably paracompact if every countable open cover of X has an open locally finite refinement. It is known that every metrizable topological space is paracompact and hence countably paracompact. It is proposed to show that in the usual metrization theorem for topological spaces [3, p. 127], the Ts condition may be replaced by the combination T2 and countably paracompactness. We will need the following lemma.