I am particularly confused with alternative formulas describing the inner and outer limits of a sequence of sets in topological spaces. The inner limit of a sequence of sets $\left\{C_n\right\}_{n\in\mathbb{N}}$ is defined as:
$$ \liminf_n C_n := \left\{x|\ x\in C_n \text{ ultimately for all } n\right\} $$
or equivalently
$$ \liminf_n C_n = \left\{x|\exists N_0\in\mathbb{N} \text{ such that }x\in C_k,\ \forall k\geq N_0\right\} $$
Then on one hand we can prove the following well known result:
Proposition 1 : The inner limit of a sequence of sets is: $$ \liminf_n C_n = \bigcup_{n=1}^{\infty}\bigcap_{m=n}^{\infty} C_m \ \ \text{(1)} $$ Proof. The proof can be found in http://planetmath.org/limitofsequenceofsets (Most books of analysis coping with sequences of sets pertain to this result).
$\square$
But on the other hand we can prove that:
Proposition 2 : Let $\left(\mathcal{X},\mathcal{T}\right)$ be a Hausdorff topological space and $\left\{C_n\right\}_{n\in\mathbb{N}}$ be a sequence of sets in $\mathcal{X}$. Then, $$ \liminf_n C_n = \bigcap\left\{ \overline{\bigcup_{n\in N} C_n},\ N \text{ is a cofinal subset of } \mathbb{N} \right\} $$ Note (Notation) : $\bar{C}$ stands for the closure of the set $C$.
Proof. This result can be found in the Book of R.T. Rockafellar and R.J-B. Wets, Variational Analysis (Springer, Dordrecht 2000, ISBN: 978-3-540-62772-2) as an exercise while the proof can be found in the book of G.Beer, Topologies on Closed and Convex Closed Sets (Kluwer Academic Publishers, Dordecht 1993, ISBN: 0-7923-2531-1).
$\square$
Using Proposition 1, we arrive at the following result:
Proposition 3A : Let $\left\{C_n\right\}_n$ be a decreasing sequence of sets (i.e. $C_k \subseteq C_{k+1}$ for all $k\in\mathbb{N}$). Then:
$$ \liminf_n C_n = \bigcap_{n\in\mathbb{N}}C_n $$
Let $\left\{C_n\right\}_n$ be an increasing sequence of sets (i.e. $C_k \supseteq C_{k+1}$ for all $k\in\mathbb{N}$). Then:
$$ \liminf_n C_n = \bigcup_{n\in\mathbb{N}}C_n $$
Proof. The proof can be found at http://planetmath.org/limitofsequenceofsets.
$\square$
But then, using Proposition 2 we can see that:
Proposition 3B : Let $\left\{C_n\right\}_n$ be a decreasing sequence of sets. Then:
$$ \liminf_n C_n = \bigcap_{n\in\mathbb{N}}\overline{C_n} $$
Let $\left\{C_n\right\}_n$ be a decreasing sequence of sets. Then:
$$ \liminf_n C_n = \overline{\bigcap_{n\in\mathbb{N}} C_n} $$
Note : Rockafellar and Wets give the following example which to me seems paradoxical: Consider the constant sequence $C_v=\mathbb{Q}^p$ for $v\in\mathbb{N}$ in the normed space $\left(\mathbb{R}^p,\left\| \cdot \right\| \right)$ (with the Euclidean norm). Then $\liminf_v C_v = \limsup_v C_v = \mathbb{R}^p$ (NOT $\mathbb{Q}^p$; and they underline this fact). However if we consider the element $y=\left( \pi,\pi,\ldots,\pi\right)\in \mathbb{R}^p\setminus \mathbb{Q}^p$ then for every $n\in\mathbb{N}$, $y\notin C_n$. Therefore, I would say that $y\notin \limsup_v C_v$ and $y\notin \liminf_v C_v$.