Inner Limit in Hausdorff Space by Open Neighborhoods

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Theorem

Let $\sequence {C_n}_{n \mathop \in \N}$ be a sequence of sets in a Hausdorff topological space $\struct {\XX, \tau}$.

Let $x \in \XX$.

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

Let $\NN_\infty$ denote the set of cofinite subsets of $\N$:

$\NN_\infty := \set {N \subset \N: \N \setminus N \text{ is finite} }$


Then the inner limit of $\sequence {C_n}_{n \mathop \in \N}$ is:

$\ds \liminf_n C_n = \set {x \in \XX: \forall V \in \map \mho x: \exists N \in \NN_\infty: \forall n \in N: C_n \cap V \ne \O}$

or equivalently:

$\ds \liminf_n C_n = \set {x \in \XX: \forall V \in \map \mho x: \exists N_0 \in \N: \forall n \ge N_0: C_n \cap V \ne \O}$


Proof

If $x \in \liminf_n C_n$ then there exist a sequence $\sequence {x_k}_{n \mathop \in \N}$ such that $x_k \to x$ while:

$x_k \in C_{n_k}$

and

$\sequence {n_k}_{k \mathop \in \N} \subseteq \N$ is a strictly increasing sequence of indices.

For any $V \in \map \mho x$ there exists $N_0 \in\N$ such that for all $i \ge N_0$:

$x_i \in V$

and:

$x_i \in C_{n_i}$


Thus:

$C_{n_i} \cap V \ne \O$

Therefore $x \in \set {x \in \XX: \forall V \in \map \mho x: \exists N_0 \in \N: \forall n \ge N_0: C_n \cap V \ne \O}$.

$\Box$


Let $x \in \set {x \in \XX: \forall V \in \map \mho x: \exists N \in \NN_\infty: \forall n \in N: C_n \cap V \ne \O}$.

Thus:

$\forall V \in \map \mho x: \exists N \in \NN_\infty: \forall n \in N: C_n \cap V \ne \O$

Then there exists a strictly increasing sequence:

$\sequence {n_k}_{k \mathop \in \N} \subseteq \N$

such that for every $V \in \map \mho x$:

$\exists x_k \in C_{n_k} \cap V$.

Hence $x_k \to x$ in the topology $\tau$.

$\blacksquare$


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