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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!

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I don't think Part 1 is thinking in terms of a topology on $\mathcal{P}(X)$, but of a topology on $X$; we talk about accumulation points of a subset of a topological space all the time, without giving a topology to the power set of the space. For example, saying a subset is dense is saying that the set of all accumulation points of the set is the entire space. –  Arturo Magidin Jan 13 '11 at 4:30
    
Why is my question turned into community wiki? What does it mean? –  Tim Jan 13 '11 at 6:01
    
your question turned into community wiki after you edited it more than 8 times. It is a " feature " of the underlying software. –  Willie Wong Jan 13 '11 at 17:34
    
@PantelisSopasakis: Thanks! Is the definition you quoted equivalent to any of the two definitions in my post? (1) It is not the second definition, which I mentioned in part 3, because your quote relies on topology while my second definition doesn't. In other words, if remove the closure in your quote, then it will become my second definition. (2) Is it the same as my first definition, which I quoted in Part 2 and also relies on the topology on the universal set? (3) How is $\limsup_n A_n$ is defined in your book? –  Tim Nov 22 '11 at 13:50
    
@ArturoMagidin: Pantelis recently quoted some other definition in his comment. I wonder if his quote $\lim\inf_{n} A_n := \bigcup_{n=1}^\infty\overline{\bigcap_{m=n}^\infty A_m}$ is the same as the definition in my part 2 "$x \in \liminf X_n$ 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 $k \rightarrow \infty$"? –  Tim Nov 22 '11 at 14:09

2 Answers 2

up vote 3 down vote accepted

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.

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Thanks! (1) In part 1, about a sequence of subsets that accumulate to some "accumulation subsets" of X in the quoted, it make me think of topology on the power set of X. I can imagine if it is just an informal way to say But I doubt by a "accumulation set" the article means the set of limit points of a subset of X. (2) Correct me if I am wrong: when limsup of a sequence of subsets is defined in terms of set intersection and union solely, it can be viewed as limsup of a sequence of points in a complete lattice, because the power set is a complete lattice with inclusion as the order. –  Tim Jan 13 '11 at 5:56
    
(3) I read your reply to the other question. I think you tried to explain how to view limsup of a sequence of points as limsup of a sequence of subsets there, but I would like to understand how in the opposite direction. Please see the edit to Part 4 of my original post. –  Tim Jan 13 '11 at 5:56

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 $E_n$.

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That is equivalent to the second way of definition described in my part 3. There are other ways different from this way. –  Tim Jan 13 '11 at 1:09

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