Diagonal intersection

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

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

\displaystyle\Delta_{\alpha<\delta} X_\alpha,

is defined to be

\displaystyle\{\beta<\delta\mid\beta\in \bigcap_{\alpha<\beta} X_\alpha\}.

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

\displaystyle\bigcap_{\alpha < \delta} ( [0, \alpha] \cup X_\alpha ),

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

See also

References

This article incorporates material from diagonal intersection on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

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