Inner Limit in Hausdorff Space by Open Neighborhoods

Notation
Notation 1: Let $\left(\mathcal{X},\mathcal{T}\right)$ be a topological space (not necessarily normed) and $x\in\mathcal{X}$. The set of open neighborhoods of $x$ will be hereinafter denoted as:


 * $\displaystyle \mho\left(x\right):=\left\{V\in\mathcal{T}\ ,\ x\in V\right\}$

Notation 2: The following notation will be used for the classes of cofinite and cofinal subsets of $\N$:


 * $\mathcal{N}_\infty:= \left\{N\subset \N| \mathbb{N}\setminus N \text{ is finite}\right\}$
 * $\mathcal{N}_\infty^\#:= \{N\subset \N| N \text{ is infinite}\}$

Theorem
Let $\left\{C_n\right\}_{n\in\mathbb{N}}$ be a sequence of sets in a Hausdorff topological space $\left(\mathcal{X},\mathcal{T}\right)$. Then the inner limit of $\left\{C_n\right\}_{n\in\mathbb{N}}$ is:


 * $\displaystyle \liminf_n C_n = \left\{x|\forall V\in\mho(x),\ \exists N\in \mathcal{N}_\infty, \forall n\in N: C_n\cap V\neq \emptyset\right\}$

or equivalenty:


 * $\displaystyle \liminf_n C_n = \left\{x|\forall V\in\mho(x),\ \exists N_0\in \mathbb{N},\forall n\geq N_0: C_n\cap V\neq \emptyset\right\}$

Proof
(1). If $x\in\liminf_n C_n$ then we can find a sequence $\left\{x_k\right\}_{k\in\mathbb{N}}$ such that $x_k\to x$ while $x_k\in C_{n_k}$ and $\left\{n_k\right\}_{k\in\mathbb{N}}\subseteq\mathbb{N}$ is a strictly increasing sequence of indices. For any $V\in\mho\left(x\right)$ there is a $N_0\in\mathbb{N}$ such that for all $i\geq N_0$ it is: $x_i\in V$; but also $x_i\in C_{n_i}$. Thus $C_{n_i}\cap V\neq \emptyset$. Therefore $x$ is in the right-hand side set of the equation.

(2). For the reverse direction assume that $x$ belongs to the right-hand side set of the given equation. Then, there is a strictly increasing sequence $\left\{n_k\right\}_{k\in\mathbb{N}}$. Then, for every $V\in\mho\left(x\right)$ we can find a $x_k\in C_{n_k}\cap V$. Hence, $x_k\to x$ ( in the topology $\mathcal{T}$ ).

Applications
Instead of arbitrary open sets - if $\mathcal{X}$ is a normed space - we may use open balls, i.e. sets of the form:
 * $\mathcal{B}\left(\varepsilon\right)=\{x\in\mathcal{X}:\ \|x\|<\varepsilon\}$

we denote the unit ball of $\mathcal{X}$ by $\mathcal{B}:=\mathcal{B}(1)$. Then $\mathcal{B}\left(\varepsilon\right)=\varepsilon\mathcal{B}$. This leads us to the following corollary:

Corollary : Let $\left\{C_n\right\}_{n\in\mathbb{N}}$ be a sequence of sets in a normed space $\left(\mathcal{X},\|\cdot\|\right)$. Then,
 * $\displaystyle \liminf_n C_n = \left\{x|\forall \varepsilon>0,\ \exists N\in \mathcal{N}_\infty,\forall n\in N: x\in C_n+\varepsilon\mathcal{B}\right\}$