Let $X$ be a set. Define the following notion of convergence on the power set $P(X)$: A sequence of sets $(A_i)_{i\in \mathbb{N}}$ is convergent if and only if the limsup and the liminf of the sequence are equal. Is there a topology $T$ on $P(X)$ such that $(A_i)_{i\in \mathbb{N}}$ converges if and only if $(A_i)_{i\in \mathbb{N}}$ converges with respect to $T$?
Background: Recall that the power set of $X$ is the set of all subsets of $X$. If $\{A_i\}_{i\in \mathbb{N}}$ is a sequence of points in $P(X)$, the liminf of $\{A_i\}$ is the set of points in all but finitely many $A_i$, and the limsup of $\{A_i\}$ is the set of points in infinitely many $A_i$. The notion of convergence if and only if the limsup equals the liminf behaves in many ways that suggest it is topological. For example, a nested increasing sequence of sets $A_1 \subseteq A_2 \subseteq \ldots$ converges to its union, a nested decreasing sequence $A_1 \supseteq A_2 \supseteq \ldots$ converges to its intersection, if two sequences agree on all but finitely many points then one converges iff the other converges. However, this is not sufficient to guarantee that it is indeed topological. For example, there is no topology on the space of functions from $\mathbb{R}$ to $\mathbb{R}$ that is equivelent to convergence pointwise almost everywhere with respect to Lebesgue measure.
