On Set Convergence I

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 \mathcal{X} to be a set endowed with a Hausdorff topology \tau which we will denote by \left( \mathcal{X},\tau\right). The topology of the space defines the class of open neighborhoods of points in \mathcal{X}:

Definition 1 (Open neighbourhoods). Let \left(\mathcal{X},\tau\right) be a topological space and x\in\mathcal{X}. The set of open neighbourhoods of x is defined as:

\mho\left(x\right):=\left\{V\in\tau|x\in V\right\}

The topology of \mathcal{X} governs the convergence of sequences of elements in this space.

Definition 2 (Convergence wrt a topology). A sequence \{x_n\}_{n}\subseteq\mathcal{X} is said to converge to some x\in\mathcal{X} with respect to the topology \tau if for every V\in\mho\left(x\right) there is a N_0\in\mathbb{N} such that x_k\in V for all k\geq N_0.

We now introduce the notions of cofinal and cofinite sets in partially ordered spaces.

Definition 3 (Cofinal). Let (\Lambda,\leq) be a directed set (i.e. \leq is a preorder). Then the set \Sigma\subseteq \Lambda is called a cofinal subset of \Lambda if for all \lambda\in\Lambda there exists a \sigma\in\Sigma such that \lambda \leq \sigma. We denote the cofinal subsets of \mathbb{N} by \mathcal{N}_\infty^\#.

Cofinal subsets of \mathbb{N} look like the sets of indices of subsequences: are those subsets of \mathbb{N} with infinitely many points.

We may now prove the following:

Proposition 1. \mathcal{N}_\infty^\# := \left\{ N\subseteq \mathbb{N},\ N \text{ is infinite}\right\}

Proof. (1). Let \Sigma be an infinite subset of \mathbb{N} and n\in\mathbb{N} arbitrary. The set \mathbb{N}_n=\left\{ m\in\mathbb{N},\ m\leq n\right\} is finite, so it cannot be \Sigma\subseteq \mathbb{N}_n, therefore there is a \sigma\in\Sigma such that n \leq \sigma. Thus, \Sigma is cofinal.

(2). Assume that \Sigma is a cofinal subset of \mathbb{N}. Let us assume that \Sigma is finite. Then, \Sigma has a maximal element, say N. For every \sigma\in\Sigma, N+1\nleq \sigma. Hence \Sigma is not cofinal. This contradicts our assumption, therefore \Sigma is infinite. ♣

We will now give the following definition:

Definition 4 (Cofinite subset). Let \Lambda be any set. A set \Phi\subseteq \Lambda is called a cofinite subset of \Lambda  if \Phi^c=\Lambda\setminus \Phi is finite. Hereinafter, we shall denote the class of  cofinite subsets of \mathbb{N} by \mathcal{N}_\infty.

The cofinite subset sets are the ones which eventually look like \mathbb{N} it self, that is, they can written as \Phi=\Phi_0\cup \{\kappa, \kappa+1, \kappa+2, \ldots\}, where \Phi_0 is some subset of \mathbb{N}.

We may now give the definition of the inner limit of a sequence of sets

Definition 5 (Inner limit). Let \{C_n\}_{n} be a sequence of sets in a Hausdorff topological space \left(\mathcal{X},\tau\right). The inner limit of this sequence is defined as:

\liminf_n C_n =  \left\{x \left|  {\begin{array}{c}  \exists N \in \mathcal{N}_\infty,\ \exists x_v \in C_v\\ v\in N,\ x_v \to x  \end{array} } \right. \right\}

Where the convergence x_v \to x is meant with respect to the topology \tau.

Accordingly, the outer limit of a sequence of sets is defined as:

Definition 6 (Outer limit). Let \{C_n\}_{n} be a sequence of sets in a Hausdorff topological space \left(\mathcal{X},\tau\right). The inner limit of this sequence is defined as:

\limsup_n C_n = \left\{ x \left|  {\begin{array}{c} \exists N \in \mathcal{N}_\infty^\#,\ \exists x_v \in C_v\\ v\in N,\ x_v \to x \end{array} } \right. \right\}

where, again, the convergence x_v \to x is meant with respect to the topology \tau.

Klimconst

Figure 1. The inner and the outer limits of a constant sequence of sets – each element of which is an open set U – is the closure of U.

If \mathcal{X} 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 X is a mapping d:X\times 2^X \to [0,+\infty] defined as

d\left(x,C\right):=\inf_y\left\{\|x-y\|;\ y\in C\right\}

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 \{a_n\}_{n}\subseteq \mathbb{R} is defined as:

\liminf_n a_n = \lim_{n\to\infty}\left[ \inf_{k\geq n} a_k \right]

Accordingly, the limit superior of \{a_n\}_{n} is:

\limsup_n a_n = \lim_{n\to\infty}\left[ \sup_{k\geq n} a_k \right]

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 \mathcal{X} (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 \mathcal{X} be a set and \{A_n\}_{n} be a sequence of sets in \mathcal{X}. The limit inferior of  this sequence is defined to be the set:

\mathrm{D{-}liminf}_{n\to\infty} C_n = \left\{ x \left| {\begin{array}{c} \exists N \in \mathcal{N}_\infty,\ x \in C_v\\ \forall v\in N \end{array} }\right. \right\}

Dliminf

Figure 2. The limit inferior of a sequence of sets.

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 \mathcal{X} be a set and \{A_n\}_{n} be a sequence of sets in \mathcal{X}. The limit superior of  this sequence is defined to be the set:

\mathrm{D{-}limsup}_{n\to\infty} C_n = \left\{ x \left| {\begin{array}{c} \exists N \in \mathcal{N}_\infty^\#,\ x \in C_v\\ \forall v\in N \end{array} }\right. \right\}

Dlimsup

FIgure 3. The limit superior of a sequence of sets.

The limit inferior and the limit superior are exactly the inner and the outer limits when the space \mathcal{X} is endowed with the discrete topology, i.e. the topology of the power set of \mathcal{X}, \tau=2^\mathcal{X}. In all other cases, the inner and outer limit yield quite different results that the limits inferior and superior. In all cases it holds:

\mathrm{D{-}liminf}_{n\to\infty} C_n \subseteq \liminf_{n\to\infty} C_n

Consider for example the case of \mathbb{R} with the usual topology and the sequence of sets:

C_n=\left\{ \begin{array}{c} \mathbb{Q},\ n\text{ is odd}\\ \mathbb{R}\setminus\mathbb{Q},\ n\text{ is even} \end{array} \right.

Then, \mathrm{D{-}liminf}_{n\to\infty} C_n = \varnothing while \liminf_n C_n = \mathbb{R}^n.

A well known property of \mathrm{D{-}liminf} is stated as follows:

Proposition 2. The limit inferior of a sequence of sets is:

\mathrm{D{-}liminf}_{n\to\infty} C_n = \bigcup_{n=1}^{\infty} \bigcap_{m=n}^{\infty} C_m

Proof. First, let us define B_k=\bigcap_{j\geq n}C_j. Then the right hand side of equation we want to prove is written as \bigcup_{n\in\mathbb{N}}B_n

(1). Assume that x\in C_i for all but finitely many indices i. Then, there is a M\in\mathbb{N} so that for all m\geq M it is x\in C_m. We notice that if k\geq M then x\in C_j for all j\geq k. Therefore,  x\in\bigcap_{j\geq k}C_j=B_k

If on the other hand k\leq M, then we can find \hat{k}:=\max\left\{k,M\right\} so that x\in\bigcap_{j\geq \hat{k}}C_j=B_{\hat{k}}

Thus, for arbitrary index k\in\mathbb{N}, there is always a \hat{k}\in\mathbb{N} such that x\in B_{\hat{k}} which means that x\in\bigcup_{n\in\mathbb{N}}B_n.

(2). Let us assume that  x\in \bigcup_{n=1}^\infty B_n, but there are infinitely many indices i, such that x\notin C_i. Let \{s_j\}_{j}\subseteq \mathbb{N} be a strictly increasing sequence such that x\notin C_{s_j} – note that s_j\geq j. For any k\in\mathbb{N} we have x\in C_{s_k}\supseteq \bigcap_{s_j \geq s_k}C_{s_j} \supseteq \bigcap_{j\geq k} C_j=B_k which means that x\notin B_k. This holds true for all k\in\mathbb{N}, thus x\notin \bigcup_{n=1}^\infty B_n 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:

\mathrm{D{-}limsup}_{n\to\infty} C_n = \bigcap_{n=1}^{\infty}\bigcup_{m=n}^{\infty} C_m

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

C_{\nu} = [\tfrac{1}{\nu}, \tfrac{1}{\nu+1}]

Then, by definition the limit superior \mathrm{D{-}limsup}_\nu C_\nu = \varnothing because we cannot find a sequence of integers N\subseteq \mathbb{N} so that x_{\nu} \in C_{\nu} and also – to avoid any misunderstanding – notice that 0\notin C_{\nu} for all \nu\in \mathbb{N}.

set-seq-02318Xi

Figure 4. Sequence of sets which converges to the empty set in the discrete topology (using the limit superior), but converges to {0} using the definitions of inner and outer limits.

However, this sequence converges to \{0\}; we can see that by taking an element x_n\in C_n, e.g,. the centre of each set and notice that x_{\nu}\to 0.

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2 responses

  1. […] post comes as a sequel of On Set Convergence I where we introduced some necessary notions that are useful for studying how sets converge in a […]

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  2. […] elements of that sequence and many other interesting properties. * Read first the previous posts: On Set Convergence I and […]

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