# a space $X$ is Hausdorff if and only if $\mathrm{\Delta}(X)$ is closed

###### Theorem.

A space $X$ is Hausdorff^{} if and only if

$$\{(x,x)\in X\times X\mid x\in X\}$$ |

is closed in $X\times X$ under the product topology.

###### Proof.

First, some preliminaries: Recall that the diagonal map $\mathrm{\Delta}:X\to X\times X$ is defined as $x\stackrel{\mathrm{\Delta}}{\u27fc}(x,x)$. Also recall that in a topology^{} generated by a basis (like the product topology), a set $Y$ is open if and only if, for every point $y\in Y$, there’s a basis element $B$ with $y\in B\subset Y$. Basis elements for $X\times X$ have the form $U\times V$ where $U,V$ are open sets in $X$.

Now, suppose that $X$ is Hausdorff. We’d like to show its image under $\mathrm{\Delta}$ is closed. We can do that by showing that its complement $\mathrm{\Delta}{(X)}^{c}$ is open. $\mathrm{\Delta}(X)$ consists of points with equal coordinates, so $\mathrm{\Delta}{(X)}^{c}$ consists of points $(x,y)$ with $x$ and $y$ distinct.

For any $(x,y)\in \mathrm{\Delta}{(X)}^{c}$, the Hausdorff condition gives us disjoint open $U,V\subset X$ with $x\in U,y\in V$. Then $U\times V$ is a basis element containing $(x,y)$. $U$ and $V$ have no points in common, so $U\times V$ contains nothing in the image of the diagonal map: $U\times V$ is contained in $\mathrm{\Delta}{(X)}^{c}$. So $\mathrm{\Delta}{(X)}^{c}$ is open, making $\mathrm{\Delta}(X)$ closed.

Now let’s suppose $\mathrm{\Delta}(X)$ is closed. Then $\mathrm{\Delta}{(X)}^{c}$ is open. Given any $(x,y)\in \mathrm{\Delta}{(X)}^{c}$, there’s a basis element $U\times V$ with $(x,y)\in U\times V\subset \mathrm{\Delta}{(X)}^{c}$. $U\times V$ lying in $\mathrm{\Delta}{(X)}^{c}$ implies that $U$ and $V$ are disjoint.

If we have $x\ne y$ in $X$, then $(x,y)$ is in $\mathrm{\Delta}{(X)}^{c}$. The basis element containing $(x,y)$ gives us open, disjoint $U,V$ with $x\in U,y\in V$. $X$ is Hausdorff, just like we wanted. ∎

Title | a space $X$ is Hausdorff if and only if $\mathrm{\Delta}(X)$ is closed |
---|---|

Canonical name | ASpacemathnormalXIsHausdorffIfAndOnlyIfDeltaXIsClosed |

Date of creation | 2013-03-22 14:20:47 |

Last modified on | 2013-03-22 14:20:47 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 9 |

Author | mathcam (2727) |

Entry type | Proof |

Classification | msc 54D10 |

Related topic | DiagonalEmbedding |

Related topic | T2Space |

Related topic | ProductTopology |

Related topic | SeparatedScheme |