Inner Limit in Hausdorff Space by Set Closures

Theorem
Let $\struct {\XX, \tau}$ be a Hausdorff space.

Let $\sequence {C_n}_{n \mathop \in \N}$ be a sequence of sets in $\XX$.

Then:
 * $\displaystyle \liminf_n C_n = \bigcap_{N \mathop \in \NN_\infty^\#} \map \cl {\bigcup_{n \mathop \in N} C_n}$

where:
 * $\cl$ denotes set closure
 * $\NN_\infty^\#$ denotes the set of cofinal subsets of $\N$.

Proof
$(1)$: Let:


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

Let:


 * $\Sigma \in \NN_\infty^\#$

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

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


 * $W \cap C_n \ne \O$

Thus:


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

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

Then there exists an open neighborhood of $x$.

Let $\map \mho x := \set {V \in \tau: x \in V}$ denote the set of open neighborhoods of $x$.

Let $W \in \map \mho x$ such that:
 * $\Sigma_0 := \set {n \in \N: W \cap C_n = \O}$

is cofinal.

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

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