Diagonal intersection

Diagonal intersection is a term used in mathematics, especially in set theory.

If ${\displaystyle \displaystyle \delta }$ is an ordinal number and ${\displaystyle \displaystyle \langle X_{\alpha }\mid \alpha <\delta \rangle }$ is a sequence of subsets of ${\displaystyle \displaystyle \delta }$, then the diagonal intersection, denoted by

${\displaystyle \displaystyle \Delta _{\alpha <\delta }X_{\alpha },}$

is defined to be

${\displaystyle \displaystyle \{\beta <\delta \mid \beta \in \bigcap _{\alpha <\beta }X_{\alpha }\}.}$

That is, an ordinal ${\displaystyle \displaystyle \beta }$ is in the diagonal intersection ${\displaystyle \displaystyle \Delta _{\alpha <\delta }X_{\alpha }}$ if and only if it is contained in the first ${\displaystyle \displaystyle \beta }$ members of the sequence. This is the same as

${\displaystyle \displaystyle \bigcap _{\alpha <\delta }([0,\alpha ]\cup X_{\alpha }),}$

where the closed interval from 0 to ${\displaystyle \displaystyle \alpha }$ is used to avoid restricting the range of the intersection.