# Urysohn’s lemma

A *normal space ^{}* is a topological space

^{}$X$ such that whenever $A$ and $B$ are disjoint closed subsets of $X$, then there are disjoint open subsets $U$ and $V$ of $X$ such that $A\subseteq U$ and $B\subseteq V$.

(Note that some authors include ${\mathrm{T}}_{1}$ in the definition,
which is equivalent^{} to requiring the space to be Hausdorff^{}.)

*Urysohn’s Lemma* states that $X$ is normal
if and only if
whenever $A$ and $B$ are disjoint closed subsets of $X$,
then there is a continuous function^{} $f:X\to [0,1]$
such that $f(A)\subseteq \{0\}$ and $f(B)\subseteq \{1\}$.
(Any such function is called an *Urysohn function*.)

A corollary of Urysohn’s Lemma
is that normal ${\mathrm{T}}_{1}$ (http://planetmath.org/T1Space) spaces are completely regular^{}.

Title | Urysohn’s lemma |

Canonical name | UrysohnsLemma |

Date of creation | 2013-03-22 12:12:34 |

Last modified on | 2013-03-22 12:12:34 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 12 |

Author | yark (2760) |

Entry type | Theorem |

Classification | msc 54D15 |

Related topic | HowIsNormalityAndT4DefinedInBooks |

Related topic | ApplicationsOfUrysohnsLemmaToLocallyCompactHausdorffSpaces |

Defines | Urysohn function |

Defines | normal space |

Defines | normal topological space |

Defines | normal |

Defines | normality^{} |