# Completely metrizable space

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In mathematics, a **completely metrizable space** (**complete topological space** or **topologically complete space**) is a topological space (*X*, *T*) for which there exists at least one metric *d* on *X* such that (*X*, *d*) is a complete metric space and *d* induces the topology *T*. This is equivalent to the condition that *X* is a G_{δ} in its Stone–Čech compactification β*X*.

The set of rational numbers is an example of a topological space that is metrizable but not completely metrizable.

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