General Topology – Limsup and Liminf of Sequence of Subsets

Question

I am confused when reading Wikipedia's article on limsup and liminf of a sequence of subsets of a set $X$.

1. 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"?

2. 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?

3. 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.

4. 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!

Answer 1

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.