Inner Limit in Hausdorff Space by Set Closures

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Theorem

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

Let $\left \langle{C_n}\right \rangle_{n \mathop \in \N}$ 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.

$\blacksquare$


Also see