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:
- $\ds \liminf_n C_n = \bigcap_{N \mathop \in \NN_\infty^\#} \map \cl {\bigcup_{n \mathop \in N} C_n}$
where:
- $\cl$ denotes set closure
- $\NN_\infty^\#$ denotes the set of cofinal subsets of $\N$.
Proof
$(1)$: Let:
- $\ds x \in \liminf_n \ C_n$
Let:
- $\Sigma \in \NN_\infty^\#$
Let $W$ be an open 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:
- $\ds x \in \map \cl {\bigcup_{n \mathop \in \Sigma} C_n}$
$(2)$: Let:
- $\ds x \notin \liminf_n C_n$
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:
- $\ds x \notin \map \cl {\bigcup_{n \mathop \in \Sigma_0} C_n}$
This completes the proof.
$\blacksquare$
Also see
- Inner Limit is Closed Set: a corollary of this theorem