Diagonal intersection

Diagonal intersection is a term used in mathematics

If

$\langle S_{i}\rangle _{i<\alpha }\,$

is a sequence then the diagonal intersection,

$\Delta _{i<\alpha }S_{i}\,$

is defined to be

$\{\beta <\alpha \mid \beta \in \bigcap _{\gamma <\beta }S_{\gamma }\}\,.$

That is, $\beta \,$ is in $\Delta _{i<\alpha }S_{i}\,$ if it is contained in the first $\beta \,$ members of the sequence. This is the same as

$\bigcap _{\gamma <\alpha }([0,\gamma ]\cup S_{\gamma })\,$

where the closed interval from 0 to γ is used to avoid restricting the range of the intersection.

See also

• Fodor's lemma
• Club set
• Club filter

diagonal intersection at PlanetMath.