Let be a separable, metric space, , and let be countable and dense. For each , choose if such intersection is nonempty. Then, is a countable subset of .

Let . Then choose . Now, choose such that . Then, , so there exists , such that

Now . So . Hence, the closure of in is equal to .

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More in general, tuo can extend a non-separable space X by adding a monadic set including a single element y, with y non-belonging to X; then in such extended space Xy you impose that the closure of such added monadic set is the whole space Xy (it is trivially verified that in this way yhe original topology in X is correctly obtained as sub-space topology from Xy, i.e. X is an authentic topological subspace of a topological “super-space” Xy). Tautologically Xy will result as the closure of a set with cardinality=1