limsup and liminf of a sequence of subsets of a set 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!
 A: First, you might also want to take a look at this answer to a similar question.
Okay: the first description assumes that there is some sort of notion of "accumulation point" at work in the set $X$, as you surmise; this may be derived from a topology. 
The second description talks about limit points, but you can apply it to any set by endowing the set with the discrete topology (every subset is open, every subset is closed). If you do that, then the definition is the usual definition of limit superior of a sequence of sets: it is the collection of all points that are in infinitely many of the terms of the sequence, while the limit inferior is the collection of all points that are in all sufficiently large terms of the sequence. 
The "second way" of defining it is in terms of unions and intersection. If $\{X_n\}_{n\in\mathbb{N}}$ is a family of sets, then
\begin{align*}
\limsup_{n\in\mathbb{N}} X_n &= \bigcap_{n=1}^{\infty}\left(\bigcup_{j=n}^{\infty} X_j\right)\\\
\liminf_{n\in\mathbb{N}} X_n &= \bigcup_{n=1}^{\infty}\left(\bigcap_{j=n}^{\infty} X_j\right).
\end{align*}
This coincides with the notion of the limit superior being the set of all limit points of infinitely many terms in the sequence, under the discrete topology; and the limit inferior being the set of all limit points of all sufficiently large-indexed terms of the sequence (again, under the discrete topology).
The notion of "accumulation point" in the first description is more informal. If you are working with a topological space, then it is limit points as described above and by "accumulation set" you should read "set of all limit points".
For your third point, in order to be able to talk about joins and meets you need to have some sort of complete lattice order on your set, so that you can talk about those infinite meets and infinite joins; this is the case, for instance, in the real numbers; appropriately interpreted, you do get essentially the definition you propose, though you need to tweak it a bit in order to actually get what the actual definition is (see the other answer quoted above); you don't actually work with the points themselves, but with a slightly different set determined by the points. 
I think that the previous answer linked to answers essentially your fourth point, of how to interpret limit superior and limit inferior of a sequence of points as a special case of limit superior and limit inferior of sets; but if this is not the case, point it out and I'll try to answer it de nuovo. 
A: I always found the following definitions of superior limits and inferior limits intuitive: Let $\{E_n \}$ be a sequence of sets ($n = 1,2, \dots$). The superior limit of $\{E_n \}$ is the set consisting of those points which belong to infinitely many $E_n$. The inferior limit is the set of all those points that belong to all but a finite number of the $E_n$. 
