The Baire space II – the Baire Category Theorem

Having established the Baire space, we will here state and prove a version of the Baire Category Theorem, which is an important tool used within functional analysis, as well as also having applications in other parts of analysis and topology. After proving the theorem, we will state a few results in which the proofs rely on the theorem, to get a feeling of its power. Continue reading

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The Baire space I – getting off the ground

An area in mathematics concerning a special kind of topological spaces called Baire spaces is an example of a notion that intersects three branches of mathematics, namely topology, analysis and set theory – this intersection in particular is called descriptive set theory. We start off by defining the space, then exploring its properties and ending up looking at its applications throughout analysis and topology. Continue reading