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
Fodor's lemma
Club set
Club filter
References
Thomas Jech, Set Theory, The Third Millennium Edition, Springer-Verlag Berlin Heidelberg New York, 2003, page 92.
Akihiro Kanamori, The Higher Infinite, Second Edition, Springer-Verlag Berlin Heidelberg, 2009, page 2.
This article incorporates material from diagonal intersection on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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