View source for The diagonal (set)

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==Definition==
Let {{M|X}} be a [[set]] and let {{M|X\times X}} denote the 2-fold [[Cartesian product]] of {{M|X}} with itself, we call the following set the ''diagonal'' of {{M|X}} in {{M|X\times X}}{{rHTJWV}}:
* {{M|D:\eq\{(x,x)\in X\times X\ \vert\ x\in X\} }}
** Here {{M|(x,x)}} denotes the [[ordered pair]] whose first element is {{M|x}} and whose second element is also {{M|x}}
==Uses==
In topology:
* [[A topological space is Hausdorff if and only if the diagonal of the space is closed in its 2-fold Cartesian product considered with the product topology|A topological space {{M|(X,\mathcal{J})}} is Hausdorff {{M|\iff}} the diagonal {{M|D\subseteq X\times X}} is closed in {{M|X\times X}} (considered with its product topology)]]
==References==
<references/>
{{Definition|Set Theory}}