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Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 2 / 11F / ii / Solution

Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 2 / 11F / i / Solution

Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 4 / 22F / Solution

Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 2 / 22G / Solution

Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 4 / 2G / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 4 / 2G / b / Solution

Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 8 / 1 / Solution

Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 2 / 22F / Solution

Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 3 / 2F / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 3 / 2F / b / Solution

Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 1 / 22H / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 1 / 22H / c / Solution

Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 8 / 1 / Solution

Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 1 / 22H / Solution

Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 1 / 2F / ii / Solution

Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 1 / 2F / i / Solution

Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 1 / 2F / ii / Solution

Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 1 / 2F / i / Solution

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Every complete metric space is a Baire space: every countable intersection of open dense subsets is dense. Equivalently, no nonempty open subset is a countable union of nowhere-dense subsets.

To prove the dense-intersection form, let $V$ be nonempty and open and let $G_1,G_2,\ldots$ be open dense sets. Successively choose closed balls
$$
\overline B(x_n,r_n)\subset B(x_{n-1},r_{n-1})\cap G_n,
\qquad 0<r_n<2^{-n},
$$
starting with a ball contained in $V\cap G_1$. The centres form a Cauchy sequence. Its limit belongs to every closed ball, and hence to $V\cap\bigcap_nG_n$.

Baire category theorem

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 Incoming links: Baire category theorem