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 $\N$:


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

Theorem
Let $\left \langle{C_n}\right \rangle_{n \in \N}$ be a sequence of sets in a Hausdorff topological space $\left(\mathcal{X},\mathcal{T}\right)$.

Then the inner limit of $\left \langle{C_n}\right \rangle_{n \in \N}$ is:


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

or equivalently:


 * $\displaystyle \liminf_n C_n = \left\{{x: \forall V \in \mho(x): \exists N_0 \in \N: \forall n \geq N_0: C_n \cap V \ne \varnothing}\right\}$

Proof
(1). If $x\in\liminf_n \ C_n$ then we can find a sequence $\left \langle{x_k}\right \rangle_{n \in \N}$ such that


 * $x_k\to x$ while $x_k\in C_{n_k}$

and


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

For any $V\in\mho\left(x\right)$ there is a $N_0\in\N$ 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. That is,


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

Then, there is a strictly increasing sequence $\left\{n_k\right\}_{k\in\N}$ such that 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 $\left \langle{C_n}\right \rangle_{n \in \N}$ be a sequence of sets in a normed space $\left(\mathcal{X},\|\cdot\|\right)$. Then,
 * $\displaystyle \liminf_n C_n = \left\{x|\forall \varepsilon>0,\ \exists N\in \mathcal{N}_\infty,\forall n\in N: x\in C_n+\varepsilon\mathcal{B}\right\}$