We give the definitions of inner and outer limits for sequences of sets in topological and normed spaces and we provide some important facts on set convergence on topological and normed spaces. We juxtapose the notions of the limit superior and limit inferior for sequences of sets and we outline some facts regarding the Painlevé-Kuratowski convergence of set sequences.
This article will be in three parts. Here we provide some basic definitions and facts which we shall later use to provide a topological characterisation of set convergence and study some interesting properties of the Painlevé-Kuratowski convergence.
In what follows we always consider to be a set endowed with a Hausdorff topology which we will denote by . The topology of the space defines the class of open neighborhoods of points in :
Definition 1 (Open neighbourhoods). Let be a topological space and . The set of open neighbourhoods of is defined as:
The topology of governs the convergence of sequences of elements in this space.
Definition 2 (Convergence wrt a topology). A sequence is said to converge to some with respect to the topology if for every there is a such that for all .
We now introduce the notions of cofinal and cofinite sets in partially ordered spaces.
Definition 3 (Cofinal). Let be a directed set (i.e. is a preorder). Then the set is called a cofinal subset of if for all there exists a such that . We denote the cofinal subsets of by .
Cofinal subsets of look like the sets of indices of subsequences: are those subsets of with infinitely many points.
We may now prove the following:
Proof. (1). Let be an infinite subset of and arbitrary. The set is finite, so it cannot be , therefore there is a such that . Thus, is cofinal.
(2). Assume that is a cofinal subset of . Let us assume that is finite. Then, has a maximal element, say . For every , . Hence is not cofinal. This contradicts our assumption, therefore is infinite. ♣
We will now give the following definition:
Definition 4 (Cofinite subset). Let be any set. A set is called a cofinite subset of if is finite. Hereinafter, we shall denote the class of cofinite subsets of by .
The cofinite subset sets are the ones which eventually look like it self, that is, they can written as , where is some subset of
We may now give the definition of the inner limit of a sequence of sets
Definition 5 (Inner limit). Let be a sequence of sets in a Hausdorff topological space . The inner limit of this sequence is defined as:
Where the convergence is meant with respect to the topology .
Accordingly, the outer limit of a sequence of sets is defined as:
Definition 6 (Outer limit). Let be a sequence of sets in a Hausdorff topological space . The inner limit of this sequence is defined as:
where, again, the convergence is meant with respect to the topology .
If is a normed space then specific conclusions can be drawn exploiting the well known properties of the norm and the norm-balls. We introduce the notion of the point-to-set distance mapping.
Definition 7 (Point-to-set distance). The point-to-set distance on is a mapping defined as
The limit inferior – which should not be confused with the inner limit – and the limit superior – which, again, is not the same as the outer limit – of a sequence of real numbers will be of high importance in what follows. We give the following definition:
Definition 8 (Limit inferior, limit superior). The limit inferior of a sequence is defined as:
Accordingly, the limit superior of is:
Notice that the above definition does not assume the existence of a topology. The limit inferior of a sequence of elements or subsets of a space (which does not need to be endowed with any topology) is given as follows
Definition 9.a (Limit inferior of a seq. of sets). Let be a set and be a sequence of sets in . The limit inferior of this sequence is defined to be the set:
What is the same, we may define the limit superior of a sequence of sets as:
Definition 9.b (Limit superior of a seq. of sets). Let be a set and be a sequence of sets in . The limit superior of this sequence is defined to be the set:
The limit inferior and the limit superior are exactly the inner and the outer limits when the space is endowed with the discrete topology, i.e. the topology of the power set of , . In all other cases, the inner and outer limit yield quite different results that the limits inferior and superior. In all cases it holds:
Consider for example the case of with the usual topology and the sequence of sets:
Then, while .
A well known property of is stated as follows:
Proposition 2. The limit inferior of a sequence of sets is:
Proof. First, let us define . Then the right hand side of equation we want to prove is written as
(1). Assume that for all but finitely many indices i. Then, there is a so that for all it is . We notice that if then for all . Therefore,
If on the other hand , then we can find so that
Thus, for arbitrary index , there is always a such that which means that .
(2). Let us assume that but there are infinitely many indices , such that . Let be a strictly increasing sequence such that – note that . For any we have which means that . This holds true for all , thus which contradicts our initial assumption. This completes the proof. ♣
In a similar fashion we can prove the following fact regarding the limit superior:
Proposition 3. The limit superior of a sequence of sets is:
The proof of proposition 3 goes along the same lines of proposition 2. We mentioned the notions of the limits superior and inferior which are not topological notions for the sake of completeness. These notions are encountered in measure theory, but are of little use in convex and variational analysis.
We will close this post with an example of a sequence of sets
Then, by definition the limit superior because we cannot find a sequence of integers so that and also – to avoid any misunderstanding – notice that for all .
However, this sequence converges to ; we can see that by taking an element , e.g,. the centre of each set and notice that .