Intuition behind the diagonal intersection - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T16:19:57Z http://mathoverflow.net/feeds/question/107179 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/107179/intuition-behind-the-diagonal-intersection Intuition behind the diagonal intersection Asaf Karagila 2012-09-14T13:18:11Z 2012-09-14T17:46:17Z <p>Suppose that for all $\alpha&lt;\kappa$ we have that $A_\alpha\subseteq\kappa$. We define the diagonal intersection to be $$\bigtriangleup_{\alpha&lt;\kappa}A_\alpha = \left\lbrace\xi&lt;\kappa\ \middle|\ \xi\in\bigcap_{i&lt;\xi}A_i\right\rbrace$$</p> <p>One of the most surprising theorems in basic set theory, I think, is that if $A_\alpha$ is closed and unbounded (and $\kappa$ is regular and uncountable) then this diagonal intersection is also a closed and unbounded set.</p> <p>Looking at it from a measure theoretic point of view now, clubs correspond to sets of measure one. Is there any measure theoretic operation which corresponds to diagonal intersections?</p> <p>Are there possibly other analogies in mathematics which can be used to describe this construction in a rather simple way that non-set theorists could relate to?</p> <p>Furthermore, it is quite clear that changing the order of the $A_\alpha$ or taking a subsequence can completely change the resulting set. Is there some invariance? For example, up to order the result is unique modulo a non-stationary set?</p> http://mathoverflow.net/questions/107179/intuition-behind-the-diagonal-intersection/107181#107181 Answer by Sean Cox for Intuition behind the diagonal intersection Sean Cox 2012-09-14T13:52:53Z 2012-09-14T13:52:53Z <p>The diagonal intersection corresponds to the infimum in the boolean algebra $\mathbb{B}_I:=P(\kappa)/I$ (where $I$ is the nonstationary ideal, or more generally where $I$ is any normal ideal on $\kappa$). More precisely: if $Z \subset P(\kappa)/I$ and $|Z| = \kappa$, then $Z$ has an infimum in $\mathbb{B}_I$, and this infimum is exactly the'' diagonal intersection of representatives from the members of $Z$. (This diagonal intersection does not depend on the particular $\kappa$-enumeration of $Z$ or the choice of representatives from the equivalence classes in $Z$; they'll all yield the same element of $\mathbb{B}_I$).</p>