So, I'm currently taking a course in set theory, and we've been using diagonal intersections and diagonal unions. I know the definitions (namely, if $(X_\xi:\xi<\kappa)$ is a sequence of subsets of $\kappa$ for some regular uncountable $\kappa$, then $\alpha\in\Delta_{\xi<\kappa}X_\xi\Longleftrightarrow\alpha\in\bigcap_{\xi<\alpha}X_\xi$ for diagonal intersection).
But, I don't entirely know how to "visualize" this or build an intuition for it. Like, why would this be an important concept to have? Why should we care about filters that are closed under diagonal intersections, and why should these be, in some sense, "normal?"