Inner Limit in Hausdorff Space by Set Closures

Theorem
Let $\left({\mathcal X, \tau}\right)$ be a Hausdorff topological space and $\left \langle{C_n}\right \rangle_{n \in \N}$ be a sequence of sets in $\mathcal{X}$.

Then,


 * $\displaystyle \liminf_n \ C_n = \bigcap_{N \mathop \in \mathcal N_\infty^\#} \operatorname{cl} \bigcup_{n \mathop \in N} C_n$

where $\operatorname{cl}$ stands for the closure of a set and $\mathcal N_\infty^\#$ stands for the family of cofinal subsets of $\N$.

Proof
$(1)$: Assume that:


 * $\displaystyle x \in \liminf_n \ C_n$

and let:


 * $\Sigma \in \mathcal N_\infty^\#$

Let $W$ be a neighborhood of $x$.

Then there is a $N_0 \in \N$ such that for all $n \ge N_0$ such that $n \in \Sigma$:


 * $W \cap C_n \ne \varnothing$

Thus:


 * $\displaystyle x \in \operatorname{cl} \bigcup_{n \mathop \in \Sigma} C_n$

$(2)$: Assume that


 * $\displaystyle x \notin \liminf_n \ C_n$

Then there is an open neighborhood of $x$, let $W \in \mho \left({x}\right)$, such that the set:


 * $\Sigma_0 := \left\{{n \in \N: W \cap C_n = \varnothing}\right\}$

is cofinal.

Therefore:


 * $\displaystyle x \notin \operatorname{cl} \bigcup_{n \mathop \in \Sigma_0} C_n$

This completes the proof.

Also see

 * Inner Limit is Closed Set: a corollary of this theorem


 * Inner Limit in Hausdorff Space by Open Neighborhoods
 * Inner Limit of Sequence of Sets in Normed Space