mod09lec52 - The Baire Category Theorem

We begin the proof of the Baire Category theorem, first proving it for compact Hausdorff spaces. For this purpose,we give a Separation Lemma for compact Hausdorff spaces, which allows us to separate a point and a disjoint closed set with open neighbourhoods. We then explain the various steps in the Baire Category theorem, using the finite intersection property of compact spaces to conclude the proof.