i came across a concept on set theory in a book on topology and i can't seem to understand it.
if $U$ is the underlying universal set and if $\left\{A_{i}\right\}$ is a class of subsets of $U$ for an indexed set $I=\left\{1,2,3,\ldots \right\}$ ,then if $\left\{A_{i}\right\}$ is an empty class then it follows that
i don't understand how this holds true.vacuous definitions are really difficult to grasp and digest.please help

  • 1
    $\begingroup$ The idea is that the intersection is the collection of things in every $A_i$, if there aren't any $A_i$s then every element of $U$ is in every $A_i$ (trivially, as there aren't any). Similarly convention dictates that the empty union is the empty set, because an element is in the union iff it is in at least one of the things being unioned. $\endgroup$ – James Aug 19 '15 at 19:25
  • $\begingroup$ Arbitrary Union: $\bigcup_{\lambda\in\Lambda}A_\lambda:=\{x\in X:\exists\lambda\in\Lambda:x\in A_\lambda\}$ Arbitrary Intersection: $\bigcap_{\lambda\in\Lambda}A_\lambda:=\{x\in X:\forall\lambda\in\Lambda:x\in A_\lambda\}$ Empty Family: $\Lambda=\{\;\}:\quad\bigcap_{\lambda\in\Lambda}A_\lambda=\varnothing\quad \bigcap_{\lambda\in\Lambda}A_\lambda=X$ $\endgroup$ – C-Star-W-Star Aug 19 '15 at 19:30