Inner Limit in Hausdorff Space by Open Neighborhoods

Notation

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

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

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 $\left(\mathcal{X},\tau\right)$.

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

$\left \langle{n_k}\right \rangle_{k\in\N}\subseteq\N$ 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 $\left \langle{n_k}\right \rangle_{k\in\N}\subseteq\N$ 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 $\tau$ ).

$\blacksquare$

See also

• Inner Limit of Sequence of Sets in Normed Space described via the point-to-set distance function induced by the norm of the space
• Inner Limit in Normed Spaces by Open Balls
• Inner Limit in Hausdorff Space by Set Closures
• Inner Limit is a Closed Set