I am confused when reading Wikipedia's article on limsup and liminf of a sequence of subsets of a set $X$.
It says there are two different ways to define them, but first gives what is common for the two. Quoted:
There are two common ways to define the limit of sequences of set. In both cases:
The sequence accumulates around sets of points rather than single points themselves. That is, because each element of the sequence is itself a set, there exist accumulation sets that are somehow nearby to infinitely many elements of the sequence.
The supremum/superior/outer limit is a set that joins these accumulation sets together. That is, it is the union of all of the accumulation sets. When ordering by set inclusion, the supremum limit is the least upper bound on the set of accumulation points because it contains each of them. Hence, it is the supremum of the limit points.
The infimum/inferior/inner limit is a set where all of these accumulation sets meet. That is, it is the intersection of all of the accumulation sets. When ordering by set inclusion, the infimum limit is the greatest lower bound on the set of accumulation points because it is contained in each of them. Hence, it is the infimum of the limit points.
The difference between the two definitions involves the topology (i.e., how to quantify separation) is defined. In fact, the second definition is identical to the first when the discrete metric is used to induce the topology on $X$.
Because it mentions that a sequence of subsets of a set $X$ accumulate to some accumulation subsets of $X$, are there some topology on the power set of the set for this accumulation to make sense? What kind of topology is that? Is it induced from some structure on the set $X$? Is it possible to use mathematic symbols to formalize what it means by "supremum/superior/outer limit" and "infimum/inferior/inner limit"?
If I understand correctly, here is the first way to define limsup/liminf of a sequence of subsets. Quoted:
General set convergence
In this case, a sequence of sets approaches a limiting set when its elements of each member of the sequence approach that elements of the limiting set. In particular, if $\{X_n\}$ is a sequence of subsets of $X$, then:
$\limsup X_n$, 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, $x \in \limsup X_n$ if and only if there exists a sequence of points $x_k$ and a subsequence $\{X_{n_k}\}$ of $\{X_n\}$ such that $x_k \in X_{n_k}$ and $x_k \rightarrow x$ as $k \rightarrow \infty$.
$\liminf X_n$, which is also called the inner limit, consists of those elements which are limits of points in $X_n$ for all but finitely many n (i.e., cofinitely many n). That is, $x \in \liminf X_n$ if and only if there exists a sequence of points $\{x_k\}$ such that $x_k \in X_k$ and $x_k \rightarrow x$ as $k \rightarrow \infty$.
So I think for this definition, $X$ is required to be a topological space. This definition is expressed in terms of convergence of a sequence of points in $X$ with respect to the topology of $X$. If referring back to what is common for the two ways of definitions, I will be wondering how to explain what is a "accumulation set" in this definition here and what topology the "accumulation set" is with respect to? i.e. how can the definition here fit into aforementioned what is common for the two ways?
It says there are two ways to define the limit of a sequence of subsets of a set $X$. But there seems to be just one in the article, as quoted in 2. So I was wondering what is the second way it refers to?
As you might give your answer, here is my thought/guess (which has actually been written in the article but not in a way saying it is the second one). Please correct me.
In an arbitrary complete lattice, by viewing meet as inf and join as sup, the limsup of a sequence of points $\{x_n\}$ is defined as: $\limsup \, x_n = \inf_{n \geq 0} \left(\sup_{m \geq n} \, x_m\right) = \mathop{\wedge}\limits_{n \geq 0}\left( \mathop{\vee}\limits_{m\ \geq n} \, x_m\right) $ similarly define liminf.
The power set of any set is a complete lattice with union and intersection being join and meet, so the liminf and limsup of a sequence of subsets can be defined in the same way. I was wondering if this is the other way the article tries to introduce? If it is, then this second way of definition does not requires $X$ to be a topological space. So how can this second way fits to what is common for the two ways in Part 1, which seems to requires some kind of topology on the power set of $X$?
I understand this way of definition can be shown to be equivalent to a special case of the first way in my part 2 when the topology on $X$ is induced by discrete metric. This is another reason that let me doubt it is the second way, because I guess the second way should at least not be equivalent to a special case of the first way.
Can the two ways of definition fit into any definition for the general cases? In the general cases, limsup/liminf is defined for a sequence of points in a set with some structure. Can limsup/liminf of a sequence of subsets of a set be viewed as limsup/liminf of a sequence of "points". If not, so in some cases, a sequence of subsets must be treated just as a sequence of subsets, but not as a sequence of "points"?
EDIT: @Arturo: In the last part of your reply to another question, did you try to explain how limsup/liminf of a sequence of points can be viewed as limsup/liminf of a sequence of subsets? I actually want to understand in the opposite direction:
Here is a post with my current knowledge about limsup/liminf of a sequence of points in a set. For limsup/liminf of a sequence of subsets of any set $X$, defined in terms of union and intersection of subsets of $X$ as in part 3, it can be viewed as limsup/liminf of a sequence of points in a complete lattice, by viewing the power set of $X$ as a complete lattice. But for limsup/liminf of a sequence of subsets of any set defined in part 2 when X is a topological space, I was wondering if there is some way to view it as limsup/liminf of a sequence of points in some set?
It is also great if you have other approaches to understand all the ways of defining limsup/liminf of a sequence of subsets, other than the approach in Wikipedia.
Thanks and regards!