Where in this, third and final, post on the topic of set convergence we introduce the notion of Painleve-Kuratowski convergence of sequences of sets in topological spaces and we discuss about the unique properties of this notion of converges. We see that the K-limit of a constant sequence may not be equal to the elements of that sequence and many other interesting properties.

* Read first the previous posts: On Set Convergence I and II.

A sequence of sets is said to converge in the Painlené-Kuratowski sense if . This common limit, whenever it exists, will be denoted by or simply .

Since we already know that for any sequence it is , in order to show that exists and equals some set , it suffices to show that:

As it follows from the closedness properties of the inner and the outer limit, the Kuratowski limit (whenever it exists) is a closed set. Later we will give conditions under which a closed set satisfies an inclusion as in the inclusion above.

As a first example of a K-convergent sequence of sets we prove the following:

**Proposition 11 (Sequences of balls).** Let be a sequence in such that and and with . Then ,

**Proof.** It would be a waste of time to calculate the inner limit and the outer limit separately and then corroborate that the both equal the closed ball . Instead, it suffices to show that the following inclusions hold:

where .

(1). First let us note that the inner limit in this case is written as:

Assume that , or what is the same that .

For every there is a such that for all .

This means that for all ,

But also , hence there is a such that for all , it is . So, for we have:

Then, .

This way we have proved that .

(2). The second step is to prove that the outer limit also converges to . For that we use the fact that

This suggests that if then for all , there exists a strictly increasing sequence of integers such that for all . It takes similar actions as in (1) to complete the proof. ♣

Under the same assumptions, the limit inferior of this sequence of open balls converges to an *open* ball; for example it is:

A sequence of balls whose radii diverge to converges (in the Painlevé-Kuratowski sense) to the whole space as stated in the following proposition:

**Proposition 12 (Divergent balls).** Let be a sequence in such that and and with . Then,

and

.

It is quite easy to verify the above example. A similar example refers to sequences of convex polytopes.

**Proposition 13 (Convex hulls).** Let be a sequence of a set of points in a space such that for . Then .

Furthermore, limits of nested sequences of sets, either increasing or decreasing, are

particularly easy to calculate.

**Proposition 14 (Nested sequences).** Let be a nested and increasing sequence of sets. Then it is convergent in the Painlevé-Kuratowski sense and .

**Proof.** Since , if is a cofinal subset of , then and the equality

also holds for their closures. Therefore,

Similarly, we carry out the calculation for the from which it follows that

And this completes the proof. ♣

The Painlevé-Kuratowski convergence can be described using set inclusions which make it easy to check whether the K-limit of a given sequence of sets exists. Since for any sequence of sets it is then it will be whenever:

It is therefore expedient to study under what conditions a given set is inside or is a superset of . Some first results are given in the following theorem:

**Theorem 1. **Let be a sequence of sets in a Hausdorff topological space and be a closed set. Then it is if and only if for every such that , there exists a such that for all .

The following theorem provides conditions for the inclusion to hold.

**Theorem 2 (Hit-and-miss criteria).** [1, Thm. 4.5] For and a *closed* set, (a) if and only if for every compact set with , there exists so that for all . (b) if and only if for every open set , there is a cofinite set such that for all .

In normed spaces, the above stated results can be restated using open balls instead of arbitrary open sets and closed balls instead of arbitrary compact sets. If the space is additionally first countable (every local topological basis has a countable sub-basis), then we can consider a countable collection of open sets (e.g. open balls).

The following theorem provides sufficient conditions for a sequence of sets to be K-convergent:

**Theorem 3.** Let be a Hausdorff topological space and a sequence of subsets of . Let . If whenever the set is infinite, it is also cofinite, then is K-convergent.

**References**

[1] R.T. Rockafellar and R.J-B. Wets, *Variational Analysis*, Grundlehren der mathematischen Wissenshaften, vol. 317, Springer, Dordrecht 2000, ISBN: 978-3-540-62772-2.

[2] G.Beer, *Topologies on Closed and Convex Closed Sets*, Mathematics and its applications, vol. 268, Kluwer Academic Publishers, Dordecht 1993, ISBN: 0-7923-2531-1.

[3] G. Beer, *On Convergence of Closed Sets in a Metric Space and Distance Functions*, Bulletin of the Australian Mathematical Society, vol. 31, 1985, pp. 421-432.