3

Let $X$ be a topological space. I am asking about the relations and differences between the following two different types of $\limsup$ and $\liminf$ of $A_n ⊆ X, n ∈ \mathbb{N}$, a sequence of subsets of $X$.

From Wikipedia

$\limsup A_n$, which is also called the outer limit, consists of those elements which are limits of points in $X_n$ taken from (countably) infinitely many $n$. That is, $x ∈ \limsup A_n$ if and only if there exists a sequence of points $x_k$ and a subsequence $\{A_{n_k}\}$ of $\{A_n\}$ such that $x_k ∈ A_{n_k}$ and $x_k → x$ as $k → ∞$.

$\liminf A_n$, which is also called the inner limit, consists of those elements which are limits of points in $A_n$ for all but finitely many $n$ (i.e., cofinitely many $n$). That is, $x ∈ \liminf A_n$ if and only if there exists a sequence of points $\{x_k\}$ such that $x_k ∈ A_k$ and $x_k → x$ as $k → ∞$.

From Wikipedia:

the Kuratowski limit inferior (or lower closed limit) of $A_n$ as $n → ∞$ is $$ \mathop{\mathrm{Li}}_{n \to \infty} A_{n} := \left\{ x \in X \;\left|\; \begin{matrix} \mbox{for any open neighbourhood } U \mbox{ of } x, \\ U \cap A_{n} \neq \emptyset \mbox{ for large enough } n \end{matrix} \right. \right\}; $$

and the Kuratowski limit superior (or upper closed limit) of $A_n$ as $n → ∞$ is $$\mathop{\mathrm{Ls}}_{n \to \infty} A_{n} := \left\{ x \in X \;\left|\; \begin{matrix} \mbox{for any open neighbourhood } U \mbox{ of } x, \\ U \cap A_{n} \neq \emptyset \mbox{ for infinitely many } n \end{matrix} \right. \right\}. $$

My attempt so far (correct me if I am wrong):

$$\limsup_{n \to \infty} A_n \subseteq \mathop{\mathrm{Ls}}_{n \to \infty} A_{n}.$$ Proof: For any $x \in \limsup_{n \to \infty} A_n$, there exists a sequence of points $x_k$ and a subsequence $\{A_{n_k}\}$ of $\{A_n\}$ such that $x_k ∈ A_{n_k}$ and $x_k → x$ as $k → ∞$. For any open neighborhood $U$ of $x$, there exists $j \in \mathbb{N}$ such that for all $k \geq j$, $x_k \in U$. Since $x_k ∈ A_{n_k}$, $U \cap A_n \neq \emptyset$ for infinitely many $n$. So $x \in \mathop{\mathrm{Ls}}_{n \to \infty} A_{n}$.

Similarly, I can prove $$\liminf_{n \to \infty} A_n \subseteq \mathop{\mathrm{Li}}_{n \to \infty} A_{n}.$$

Questions:

  1. I wonder how to prove or disprove that $$\limsup_{n \to \infty} A_n \equiv \mathop{\mathrm{Ls}}_{n \to \infty} A_{n}$$ and $$\liminf_{n \to \infty} A_n \equiv \mathop{\mathrm{Li}}_{n \to \infty} A_{n}?$$
  2. If they are not equal, when will they be? For example, will $X$ being first-countable suffice?
  3. Can the outer/inner limit be represented in terms of operations on subsets, such as union, intersection, closure, interior, ...? I.e. in a simple way similar to the third type of $\limsup$ and $\liminf$ in my comment below?

Thanks and regards!

Martin Sleziak
  • 53,687
Tim
  • 47,382
  • For a sequence of subsets of a topological spaces, I have seen three types of $\limsup$ and $\liminf$. Besides the above two, the third one is $\limsup A_n := \cap_{i=1}^\infty \overline{\cup_{n=i}^\infty A_n}$ and $\liminf A_n := \cup_{i=1}^\infty \overline{\cap_{n=i}^\infty A_n}$. My question here is about the relation between the outer/inner limit and Kuratowski's type. This previous question is about the relation between Kuratowski type and the third type. – Tim Mar 21 '12 at 02:33
  • This previous question is about the relation between outer/inner limit and the third type, although I now think Andre's answer hasn't addressed it completely (see my last comment there), and I also hope if someone can give some insights there. – Tim Mar 21 '12 at 02:34
  • @One correction in my first comment: $\liminf A_n := \overline{\cup_{i=1}^\infty \cap_{n=i}^\infty \overline{A_n}}$ instead of $\liminf A_n := \cup_{i=1}^\infty \overline{\cap_{n=i}^\infty A_n}$. – Tim Mar 21 '12 at 02:45
  • 1
    You did not try (the same) examples (as always) based on constant or alternating sequences with values $\mathbb Q$, $\varnothing$ and $\mathbb R$ in $\mathbb R$, although these could help you clear the situation before you ask a question. – Did Mar 21 '12 at 06:45
  • @DidierPiau: Thanks as always! If the sequence is constant i.e. $A_n = A$ for all $n$ and some subset $A$, then outer/inner limit and Kuratowski limsup/liminf are all $\overline{A}$. If the sequence is alternating ${\mathbb{Q}, \emptyset, \mathbb{R}, \mathbb{Q}, \emptyset, \mathbb{R}, ...}$, then outer limit and Kuratowski limsup are both $\emptyset$, and inner limit and Kuratowski liminf are both $\mathbb{R}$. If I am right above, the examples don't tell if the two types are different and are not sufficient to show the two types are equivalent. – Tim Mar 21 '12 at 12:04

0 Answers0