mod09lec53 - Complete Metric Spaces and the Baire Category theorem - Part 1

We introduce the notion of completeness of a metric space using Cauchy sequences. To illustrate this property, we show that R^n is a complete metric space with both the Euclidean and the Square metric. We also show that R^omega, the countable product of R with itself, is also complete, and that the rationals are not complete.Finally we prove an analogous Separation Lemma for metric spaces which we then use in a similar way to prove the Baire Category theorem. We note that completeness is only used in the last step of the proof, using a useful lemma about intersection of nested closed sets with shrinking diameters.