The intersection of a set $X$ is
\[\bigcap X = \left\{ x : (\forall A \in X : x \in A) \right\}.\]This is a set by Separation Schema: $\bigcap X \subseteq A$ for any $A \in X$.
For any sets $X_1$, $\cdots$, $X_n$ ($n \geq 2$), we write
\[X_1 \cap \cdots \cap X_n = \bigcap \{X_1, \cdots, X_n\}.\]
- \[\bigcap \empt = \empt.\]
For any set $X$,
\[X \cap \empt = \empt.\]
For any set $X$,
\[X \cap X = X.\]
For any sets $X$ and $Y$,
\[X \cap Y = Y \cap X.\]
For any sets $X$, $Y$ and $Z$,
\[(X \cap Y) \cap Z = X \cap (Y \cap Z).\]
As a corollary of Proposition 1 (II), commutativity and associativity of $\cap$:
For any set $X$, $(\powerset(X), \cap)$ is an abelian semigroup.