# Limit superior

Let $(X,\tau)$ be a topological space and let $\mathcal{A}=\{A_i\}_{i=1}^{\infty}$ be a sequence of subsets of $X$. We define the limit superior of $\mathcal{A}$ (denoted by $\limsup \mathcal{A}$) by
$\limsup A_i = \{x\in X\colon$ for any $U \in \tau$ such that $x \in U, U \cap A_i\neq \emptyset$ for infinitely many $i\}$