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,

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

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 $n\geq N_0$ such that $n\in\Sigma$:

$W\cap C_n \neq \varnothing$

Thus,

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

(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:

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

This completes the proof.

$\blacksquare$

See Also

• Inner Limit is a Closed Set - A corollary of this theorem.
• Inner Limit in Hausdorff Space by Open Neighborhoods
• Inner Limit of Sequence of Sets in Normed Space