I'm considering a set valued mapping $X(t): \mathbb{R} \to \mathcal{P}(\mathbb{R}^n)$, where $\mathcal{P}$ denotes the power set. Given a paramater $t \downarrow 0$, I thought I could define the $\underset{t \downarrow 0}{\limsup} X(t)$ as
$$\underset{t \downarrow 0}{\limsup} X(t) := \lim_{t \downarrow 0}\big ( \sup \{x \in \mathbb{R}^n \ : \ x \ \text{is a limit point of } X(t) \} \big )$$
This, however, appears to be wrong. I am told that the limit should look something like $\underset{t \downarrow 0}{\limsup} X(t) = X \subset \mathbb{R}^n$ and I believe my definition gives a single limit point instead. If anyone more familiar with these definitions could help me out I would really appreciate it!
Besides indicator functions, you can define a lim sup to be $$\limsup_{t \downarrow 0} X(t)=\{x \in \mathbb{R}^n: \forall t \in \mathbb{R}^+, \exists t_1 \in (0,t) s.t. x \in X(t_1)\} $$