# proof of Baire category theorem

Let $(X,d)$ be a complete metric space, and ${U}_{k}$ a countable^{}
collection^{} of dense, open subsets. Let ${x}_{0}\in X$ and ${\u03f5}_{0}>0$ be
given. We must show that there exists a $x\in {\bigcap}_{k}{U}_{k}$ such that

$$ |

Since ${U}_{1}$ is dense and open, we may choose an ${\u03f5}_{1}>0$ and an ${x}_{1}\in {U}_{1}$ such that

$$ |

and such that the open ball^{} of
radius ${\u03f5}_{1}$ about ${x}_{1}$ lies entirely
in ${U}_{1}$. We then choose an ${\u03f5}_{2}>0$ and a ${x}_{2}\in {U}_{2}$ such that

$$ |

and such that the open ball
of radius ${\u03f5}_{2}$ about ${x}_{2}$ lies
entirely in ${U}_{2}$. We continue by induction^{}, and construct a sequence
of points ${x}_{k}\in {U}_{k}$ and positive ${\u03f5}_{k}$ such that

$$ |

and such that the open ball of radius ${\u03f5}_{k}$ lies entirely in ${U}_{k}$.

By construction, for $$ we have

$$ |

Hence the
sequence ${x}_{k},k=1,2,\mathrm{\dots}$ is Cauchy, and converges^{} by hypothesis^{}
to some $x\in X$. It is clear that for every $k$ we have

$$d(x,{x}_{k})\le {\u03f5}_{k}.$$ |

Moreover it follows that

$$ |

and hence a fortiori

$$ |

for every $k$. By construction then, $x\in {U}_{k}$ for all $k=1,2,\mathrm{\dots}$, as well. QED

Title | proof of Baire category theorem |
---|---|

Canonical name | ProofOfBaireCategoryTheorem |

Date of creation | 2013-03-22 13:06:55 |

Last modified on | 2013-03-22 13:06:55 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 10 |

Author | rmilson (146) |

Entry type | Proof |

Classification | msc 54E52 |