Inner Limit in Hausdorff Space by Set Closures

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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.

$\blacksquare$


Also see