Intersection

The set of elements common to all of the given sets.

Intersection

The intersection of sets $A$ and $B$ is

A\cap B := \{x : (x\in A)\ \land\ (x\in B)\}.

More generally, for an indexed family $\{A_i\}_{i\in I}$ , the intersection is

\bigcap_{i\in I} A_i := \{x : \forall i\in I,\ x\in A_i\}.

Intersections encode simultaneous constraints. In topology, closed sets are closed under arbitrary intersections, and limit-point definitions often involve intersections of neighborhoods.

Examples:

$\{1,2\}\cap\{2,3\}=\{2\}$ .
$(0,2)\cap(1,3)=(1,2)$ .
If $A_n := (-1/n,1/n)$ , then $\bigcap_{n=1}^\infty A_n = \{0\}$ .