 The difference between "completely metrizable space" and "complete metric space" is in the words "there exists at least one metric" in the definition of completely metrizable space, which is not the same as "there is given a metric" (the latter would yield the definition of complete metric space).Once we make the choice of the metric on a completely metrizable space (out of all the complete metrics compatible with the topology), we get a complete metric space.In other words, the category of completely metrizable spaces is a subcategory of that of topological spaces, while the category of complete metric spaces is not (instead, it is a subcategory of the category of metric spaces).