Difference between ''complete metric space'' and ''completely metrizable space''
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. Once we make the choice of the metric on a completely metrizable space, 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. Complete metrizability is a topological property while completeness is a property of the metric.
Examples
The space, the open unit interval, is not a complete metric space with its usual metric inherited from, but it is completely metrizable since it is homeomorphic to.
The space of rational numbers with the subspace topology inherited from is metrizable but not completely metrizable.
A subspace of a completely metrizable space is completely metrizable if and only if it is G-delta set| in.
A countable product of nonempty metrizable spaces is completely metrizable in the product topology if and only if each factor is completely metrizable. Hence, a product of nonempty metrizable spaces is completely metrizable if and only if at most countably many factors have more than one point and each factor is completely metrizable.
For every metrizable space there exists a completely metrizable space containing it as a dense subspace, since every metric space has a completion. In general, there are many such completely metrizable spaces, since completions of a topological space with respect to different metrics compatible with its topology can give topologically different completions.
When talking about spaces with more structure than just topology, like topological groups, the natural meaning of the words “completely metrizable” would arguably be the existence of a complete metric that is also compatible with that extra structure, in addition to inducing its topology. For abelian topological groups and topological vector spaces, “compatible with the extra structure” might mean that the metric is invariant under translations. However, no confusion can arise when talking about an abelian topological group or a topological vector space being completely metrizable: it can be proven that every abelian topological group that is completely metrizable as a topological space also admits an invariant complete metric that induces its topology. This implies e. g. that every completely metrizable topological vector space is complete. Indeed, a topological vector space is called complete iff its uniformity is complete; the uniformity induced by a translation-invariant metric that induces the topology coincides with the original uniformity.
