Inner Limit in Hausdorff Space by Set Closures

Theorem
Let $\left({\mathcal X, \tau}\right)$ be a Hausdorff space.

Let $\left \langle{C_n}\right \rangle_{n \in \N}$ 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:
 * $\operatorname{cl}$ denotes set closure
 * $\mathcal N_\infty^\#$ denotes the set of cofinal subsets of $\N$.

Proof
$(1)$: Let:


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

Let:


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

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

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


 * $W \cap C_n \ne \varnothing$

Thus:


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

$(2)$: Let:
 * $\displaystyle x \notin \liminf_n C_n$

Then there exists an open neighborhood of $x$.

Let $\mho \left({x}\right) := \left\{ {V \in \tau: x \in V}\right\}$ denote the set of open neighborhoods of $x$.

Let $W \in \mho \left({x}\right)$ such that:
 * $\Sigma_0 := \left\{{n \in \N: W \cap C_n = \varnothing}\right\}$

is cofinal.

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

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