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On Set ConvergencePantelis Sopasakis

Abstract—We give the definitions of inner and outerlimits for sequences of sets in tolopological and normedspaces and we provide some important facts on set con-vergence on topological and normed spaces. We juxtaposethe notions of the limit superior and limit inferior forsequences of sets and we outline some facts regarding thePainleve-Kuratowski convergence of set sequences.

Index Terms—Set Convergence, Inner limit, OuterLimit, Limit inferior, Limit Superior.

I. INTRODUCTION

IN what follows we always consider X to be aset endowed with a Hausdorff topology τ which

we will denote by (X , τ). The topology of the spacedefines the class of open neighborhoods of points inX :

Definition 1: Let (X , τ) be a topological spaceand x ∈ X . The set of openneighborhoods of x isdefined as:

f (x) := {V ∈ τ |x ∈ V } (1)

The topology of X governs the convergence ofsequences of elements in X .

Definition 2: A sequence 〈xn〉n∈N ⊆ X is saidto converge to some x ∈ X with respect to thetopology τ if for everh V ∈ f (x) there is a N0 ∈ Nsuch that xk ∈ V for all k ≥ N0.

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

Definition 3: Let (Λ,≤) be a directed set (i.e. ≤is a preorder). Then the set Σ ⊆ Λ is called a cofinalsubset of Λ if for all λ ∈ Λ there exists a σ ∈ Σsuch that λ ≤ σ. We denote the cofinal subsets ofN by N#

∞.Proposition 1:

N#∞ := {N ⊆ N, N is infinite}

Proof: (1). Let Σ be an infinite subset of N andn ∈ N arbitrary. The set Nn = {m ∈ N, m ≤ n} is

Pantelis Sopasakis is with the National Technical University ofAthens, 9 Heroon Polytechneiou Street, 15780 Zografou Campus,Athens, Greece. Email: chvng@mail.ntua.gr.

finite, so it cannote be Σ ⊆ Nn, therefore there is aσ ∈ Σ such that n ≤ σ. Thus, Σ is cofinal.

(2). Assume that Σ is a cofinal subset of N. Letus assume that Σ is finite. Then, Σ has a maximalelement, let N . For every σ ∈ Σ, N+1 � σ. HenceΣ is not cofinal. This contradicts our assumption,therefore Σ is infinite.

Definition 4: Let Λ be any set. A set Φ ⊆ Λ iscalled a cofinite subset of Λ if Φc = Λ \Φ is finite.Hereinafter, we shall denote the class of cofinitesubsets of N by N∞.

Definition 5 (Inner Limit): Let 〈Cn〉n∈N be a se-quence of sets in a Hausdorff topological space(X , τ). The inner limit of this sequence is definedas:

lim infn

Cn =

{x

∣∣∣∣ ∃N ∈ N∞, ∃xv ∈ Cv

v ∈ N, xv → x

}(2)

Where the convergence xv → x is meant withrespect to the topology τ .

Accordingly, the outer limit of a sequence of setsis defined as:

Definition 6 (Outer Limit): Let 〈Cn〉n∈N be a se-quence of sets in a Hausdorff topological space(X , τ). The outer limit of this sequence is definedas:

lim supn

Cn =

{x

∣∣∣∣ ∃N ∈ N#∞, ∃xv ∈ Cv

v ∈ N, xv → x

}(3)

Where the convergence xv → x is meant withrespect to the topology τ .

If X is a normed space then specific conclusionscan be drawn exploiting the well known propertiesof the norm and the norm-balls. We introduce thenotion of the point-to-set distance mapping.

Definition 7: The point-to-set distance on X is amapping d : X × 2X → [0,+∞] defined as

d (x,C) := infy{‖x− y‖; y ∈ C} (4)

The limit inferior and the limit superior of a se-quence of real numbers will be of high importancein what follows:

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Definition 8 (Inferior & Superior limit): The li-mit inferior of a sequence 〈an〉n∈N ⊆ R is definedas:

lim infn

an = limn→∞

[infk≥n

ak

](5)

Accordingly, the limit superior of 〈an〉n∈N is:

lim supn

an = limn→∞

[supk≥n

ak

](6)

The limit inferior of a sequence of elements orsubsets of a space X (which does not need to beendowed with any topology)

Definition 9: Let X be a set and 〈An〉n∈N be asequence of sets in X . The limit inferior of 〈An〉n∈Nis defined to be the set:

D-liminfn→∞

Cn =

{x

∣∣∣∣ ∃N ∈ N∞, xv ∈ Cv

∀v ∈ N

}(7)

What is the same, we may define the limit supe-rior of a sequence of sets as:

Definition 10: Let X be a set and 〈An〉n∈N be asequence of sets in X . The limit superior of 〈An〉n∈Nis defined to be the set:

D-limsupn→∞

Cn =

{x

∣∣∣∣ ∃N ∈ N#∞, xv ∈ Cv

∀v ∈ N

}(8)

The limit inferior and the limit superior are ex-actly the inner and the outer limits when the spaceX is endowed with the discrete topology, i.e. thetopology of the power set of X , τ = 2X . In all othercases, the inner and outer limit yield quite differentresults that the limits inferior and superior. In allcases it holds:

D-liminfn→∞

Cn ⊆ lim infn→∞

Cn (9)

Consider for example the case of R with the usualtopology and the sequence of sets:

Cn =

{Q, n is odd

R \Q, n is even (10)

Then,D-liminf

n→∞Cn = ∅ (11)

whilelim inf

nCn = Rn (12)

A well known property of D-liminf is stated asfollows:

Proposition 2: The limit inferior of a sequenceof sets is:

D-liminfn→∞

Cn =∞⋃n=1

∞⋂m=n

Cm (13)

Proof: First, let us define Bk =⋂

j≥nCj . Thenthe right hand side of equation (13) is written as⋃

n∈N

Bn

(1). Assume that x ∈ Ci for all but finitely manyindices i. Then, there is a M ∈ N so that for allm ≥M it is x ∈ Cm. We notice that if k ≥M thenx ∈ Cj for all j ≥ k. Therefore,

x ∈⋂j≥k

Cj = Bk

If on the other hand k ≤ M , then we can findk := max {k,M} so that

x ∈⋂j≥k

Cj = Bk

Thus, for arbitrary index k ∈ N, there is alwaysa k ∈ N such that x ∈ Bk which means that x ∈⋃

n∈NBn.(2). Let us assume that

x ∈∞⋃n=1

Bn

but there are infinitely many indices i, such thatx /∈ Ci. Let 〈sj〉j∈N ⊆ N be a strictly increasingsequence such that x /∈ Csj - note that sj ≥ j. Forany k ∈ N we have

x ∈ Csk ⊇⋂

sj≥sk

Csj ⊇⋂j≥k

Cj = Bk

which means that x /∈ Bk. This holds true for allk ∈ N, thus x /∈

⋃∞n=1 Bn 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:

D-limsupn→∞

Cn =∞⋂n=1

∞⋃m=n

Cm (14)

Proof: The proof is analogous to the one of theprevious proposition and is omitted.

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II. TOPOLOGICAL CHARACTERIZATION

Proposition 4: Let 〈Cn〉n∈N be a sequence of setsin a Hausdorff topological space (X , τ). Then,

lim infn

Cn ={x

∣∣∣∣ ∀V ∈ f (x) , ∃N ∈ N∞∀n ∈ N : Cn ∩ V 6= ∅

}(15)

or equivalenty:

lim infn

Cn ={x

∣∣∣∣ ∀V ∈ f (x) , ∃N0 ∈ N,∀n ≥ N0 : Cn ∩ V 6= ∅

}(16)

Proof: (1). If x ∈ lim infnCn then we canfind a sequence 〈xk〉k∈N such that xk → x whilexk ∈ Cnk

and 〈nk〉k∈N ⊆ N is a strictly increasingsequence of indices. For any V ∈ f (x) there is aN0 ∈ N such that for all i ≥ N0 it is: xi ∈ V ; butalso xi ∈ Cni

. Thus Cni∩ V 6= ∅. Therefore x is

in the right-hand side set of equation (15).(2). For the reverse direction assume that x be-

longs to the right-hand side set of equation (15).Then, there is a strictly increasing sequence 〈nk〉k∈Nsuch that for every V ∈ f (x) we can find axk ∈ Cnk

∩ V . Hence, xk → x (in the topologyτ ).

Or course for every result regarding the innerlimit, there is a corresponding one for its dualobject: the outer limit. Therefore, following similarsteps with the ones in the proof of proposition 4 wecan show that:

Proposition 5: Let 〈Cn〉n∈N be a sequence of setsin a Hausdorff topological space (X , τ). Then theouter limit of 〈Cn〉n∈N is:

lim supn

Cn ={x

∣∣∣∣ ∀V ∈ f (x) , ∃N ∈ N#∞,

∀n ∈ N : Cn ∩ V 6= ∅

}(17)

or equivalenty:

lim supn

Cn =x∣∣∣∣∣∣∀V ∈ f (x) ,∃ 〈nk〉k∈N ⊆ N〈nk〉k∈N ↑,∀k ∈ N :Cnk∩ V 6= ∅

(18)

Instead of arbitrary open sets - if X is a normedspace - we may use open balls, i.e. sets of the form:

B (ε) = {x : ‖x‖ < ε}

we denote the unit ball of X by B := B (1). ThenB (ε) = εB. This leads us to the following corollary:

Corollary 6: Let 〈Cn〉n∈N be a sequence of setsin a normed space (X , ‖ · ‖). Then,

lim infn

Cn =

{x|∀ε > 0, ∃N ∈ N∞,∀n ∈ N : x ∈ Cn + εB

}(19)

and

lim supn

Cn =

{x|∀ε > 0, ∃N ∈ N#

∞,∀n ∈ N : x ∈ Cn + εB

}(20)

This corollary yields one equivalent characteriza-tion for the inner limit. The predicate “∀ε > 0”translates into the intersection over all ε > 0,the requirement “∃N ∈ N#

∞” will be written asthe union over all N ∈ N∞ and then ∀n ∈ Ncorresponds to an intersection. This allows us torestate the corollary using set elementary operationsas follows:

Corollary 7: Let 〈Cn〉n∈N be a sequence of setsin a normed space (X , ‖ · ‖). Then,

lim infn

Cn =⋂ε>0

⋃N∈N∞

⋂v∈N

Cv + εB (21)

and

lim supn

Cn =⋂ε>0

⋃N∈N#

∞

⋂v∈N

Cv + εB (22)

Before we state and prove the following resultwe need to recall the following description of theclosure of a set. Firstly, if (X , τ) is a topologicalspace and C ⊆ X , its closure is defined as:

clC :=⋂{F : F ⊇ C, F c ∈ τ} (23)

Then,Proposition 8:

clC = {x : ∀V ∈ f (x) , V ∩ C 6= ∅} (24)

Proposition 9: Let (X , τ) be a Hausdorff topo-logical space and 〈Cn〉n∈N be a sequence of sets inX . Then,

lim infn

Cn =⋂

N∈N#∞

cl⋃n∈N

Cn (25)

Proof: (1). Let x ∈ lim infnCn and let Σ ∈N#∞. Let W be a neighborhood of x. There is a

N0 ∈ N sucht that for all n ≥ N0 such that n ∈ Σ:

W ∩ Cn 6= ∅ (26)

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Thus,x ∈ cl

⋃n∈Σ

Cn (27)

(2). Assume that x /∈ lim infnCn. Then, there isan open neighborhood of x, let W 3 x, suchthat Σ0 := {n ∈ N|W ∩ Cn = ∅}. Therefore, x /∈cl⋃

n∈Σ0Cn. This completes the proof.

Proposition 9 reveals a very important propertyof the inner limit:

Corollary 10: For any sequence of sets 〈Cn〉n∈Nin a Hausdorff topological space (X , τ) the limitlim infnCn is closed.

Proof: The inner limit is given as an ar-bitrary intersection of closed set lim infnCn =⋂

N∈N#∞

cl⋃

n∈N Cn which is closed.The following corollary is another immediate

consequence of proposition 9:Corollary 11: Let 〈Cn〉n∈N be a sequence of sets

in X . Then,

∞⋂n=1

Cn ⊆ cl∞⋂n=1

Cn ⊆ lim infn

Cn (28)

The following result is preliminary to what fol-lows. I put it here just because the proof has someinteresting steps. It is expedient to know beforehandthat the interior limit of a sequence of sets in normedspaces is described by:

lim infn

Cn ={x ∈ X | lim

nd (x,Cn) = 0

}(29)

Proposition 12: Let (X , ‖ · ‖) be a normed spaceand 〈Cn〉n∈N be a sequence of sets in X . If x ∈lim infnCn then lim infn d (x,Cn) = 0.

Proof: Let x ∈ lim infnCn. We will constructa sequence 〈xk〉k∈N ⊂ X and a strictly increasingsequence of indices 〈nk〉k∈N ⊆ N such that ‖x −xk‖ ≤ 1

kwhile xk ∈ Cnk

. By proposition 9, if x ∈lim infnCn, then for any Σ1 ∈ N#

∞:

x ∈ cl⋃i∈Σ1

Ci ⇒ ∃x1 ∈⋃i∈Σ1

Ci : ‖x− x1‖ < 1

Choose n1 ∈ Σ1 to be the smallest integer suchthat x ∈ Cn1 . Set Σ2 = {σ ∈ Σ1, σ > n1}. Note:Σ2 ⊆ Σ1 is cofinal for Σ1. Then,

x ∈ cl⋃i∈Σ2

Ci ⇒ ∃x2 ∈⋃i∈Σ2

Ci : ‖x− x2‖ <1

2

and take n2 ∈ Σ2 to the be smallest such that x2 ∈Cn2 . Eventually, we construct the aforementionedsequences. We now have that:

‖xk − x‖ <1

k, xk ∈ Cnk

⇒ 0 ≤ d (x,Cnk) <

1

k

Therefore we have proven that limk d (x,Cnk) = 0,

i.e. lim infn d (x,Cn) = 0.The previous result is extended to provide a

particularly important description of the notion ofthe inner limit on normed spaces. We use the point-to-set distance function to describe the inner limitof a sequence of sets:

Proposition 13: Let (X , ‖ · ‖) be a normed spaceand 〈Cn〉n∈N be a sequence of sets in X . The innerlimit of a sequence of sets is:

lim infn

Cn ={x ∈ X | lim

nd (x,Cn) = 0

}(30)

Proof: (1). We firstly need to show that for anyx ∈ X it is

lim supn

d (x,Cn) = 0

Let us assume that lim supn d (x,Cn) > 0,i.e. there exists an increasing sequence of indices〈nk〉k∈N so that d (x,Cnk

)→k a > 0. This suggeststhat there is a ε0 > 0 such that for all k ∈ None has that d (x,Cnk

) > ε0. However, accordingto proposition 9,

x ∈ cl⋃k∈N

Cnk

while

d

(x, cl

⋃k∈N

Cnk

)≥ ε0

which contradicts our initial assumption. Hence,lim supn d (x,Cn) = 0, i.e. limn d (x,Cn) = 0 andthis way we have proven that x is in the right-handside set.

(2). Conversely, assume that x in the right-handside of (30), that is limn d (x,Cn) = 0. For anyε > 0, we can find n0 ∈ N such that d (x,Cn) ≤ ε

2for all n ≥ n0. By definition, we have that

d (x,Cn) = inf {‖x− y‖, y ∈ Cn}

thus we can find a yn ∈ Cn such that

‖yn − x‖ < d (x,Cn) +ε

2= ε

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That is:

∃ yn ∈ Cn : ‖yn − x‖ < ε

Therefore, x ∈ Cn + εB from which it follows thatx ∈ lim infnCn (see corollary 6).

Note: For a sequence 〈an〉n∈N, in order to showthat lim infn an = a, it suffices to find a subsequenceof it that converges to a, i.e. it suffices to determinea strictly increasing sequence 〈nk〉k∈N ⊆ N such thatank→k a, i.e. limk ank

= a. Using this fact we canprove the following:

Proposition 14: Let (X , ‖ · ‖) be a normed spaceand 〈Cn〉n∈N be a sequence of sets in X . The outerlimit of a sequence of sets is:

lim supn

Cn ={x ∈ X | lim inf

nd (x,Cn) = 0

}(31)

III. PAINLENE-KURATOWSKI CONVERGENCE

A sequence of sets 〈Cn〉n∈N is said to convergein the Painlene-Kuratowski sense if lim infnCn =lim supnCn. This common limit, whenever it exists,will be denoted by K-limnCn or simply limnCn.

Since we already know that for any sequence〈Cn〉c∈N it is lim infnCn ⊆ lim supnCn, in orderto show that K-limnCn exists and equals some setC, it suffices to show that:

lim supn

Cn ⊆ C ⊆ lim infn

Cn (32)

As it follows from the closedness properties ofthe inner and the outer limit, the Kuratowski limit(whenever it exists) is a closed set. Later we willgive conditions under which a closed set C satisfiesan inclusion as in (32).

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

Proposition 15: Let 〈xn〉n∈N be a sequence in Rn

such that xn → x and and 〈ρn〉n∈N ⊆ [0,∞) withρn → ρ <∞. Then ,

K-limnB (xn, ρn) = clB (x, ρ) (33)

Proof: It would be a waste of time to calculatethe inner limit and the outer limit separately andthen corroborate that the both equal the closedball clB (x, ρ). Instead, it suffices to show that thefollowing inclusions hold:

lim supnB (xn, ρn)

(2)

⊆ K(1)

⊆ lim infnB (xn, ρn)

where K = clB (x, ρ).(1). First let us note that the inner limit in this

case is written as:

lim infnB (xn, ρn) ={

x

∣∣∣∣ ∀ε > 0, ∃N ∈ N∞,∀k ∈ N, z ∈ B (xk, ρk + ε)

}Assume that z ∈ clB (x, ρ), or what is the samethat ‖z − x‖ ≤ ρ. For every ε > 0 there is a N =N (ε) ∈ N such that ‖xk − x‖ < ε

2for all k ≥ N .

This means that for all k ≥ N ,

‖z − xk‖ ≤ ‖z − x‖+ ‖xk − x‖ < ρ+ε

2

But also ρk → ρ hence there is a M = M (ε) ∈ Nsuch that for all k ≥ M , it is |ρk − ρ| < ε

2. So, for

k ≥ max {N,M} we have:

‖z − xk‖ < ρk + ε⇔ z ∈ B (xk, ρk + ε)

Thus, z ∈ lim infn→∞ B (xn, ρn). This way we haveproved that clB (x, ρ) ⊆ lim infn B (xn, ρn).

(2). The second step is to prove that the outerlimit also converges to clB (x, ρ). For that we usethe fact that

lim supnB (xn, ρn) ={

x

∣∣∣∣ ∀ε > 0, ∃N ∈ N#∞,

∀k ∈ N, z ∈ B (xk, ρk + ε)

}This suggests that if z ∈ lim supn B (xn, ρn) then forall ε > 0, there exists a strictly increasing sequenceof integers 〈nk〉k∈N such that z ∈ B (xnk

, ρk + ε)for all k ∈ N. It takes similar actions as in (1) tocomplete the proof.

Under the same assumptions, the limit inferior ofthis sequence of open balls converges to an openball. For example it is:

D-liminfn→∞

B (xn, ρn) = B (x, ρ)

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

Proposition 16: Let 〈xn〉n∈N be a sequence in Rp

such that xn → x and and 〈ρn〉n∈N ⊆ [0,∞) withρn →∞. Then ,

K-limnB (xn, ρn) = Rp (34)

andK-lim

nB (xn, ρn)c = ∅ (35)

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A similar example refers to sequences of convexpolytopes.

Proposition 17: Let 〈xin〉n∈N be a sequence ofpoints in a space (X , τ) such that xin

n→ xi for i ∈ I.Then conv {xin}i

K→ cl conv {xi}.Limits of nested sequences of sets, either increas-

ing or descreasing, are particularly easy to calculate.Proposition 18: Let 〈Cn〉n∈N be a nested and

increasing sequence of sets. Then it is convergentin the Painlene-Kuratowski sense and K-limnCn =cl⋃

n∈NCn.Proof: Since C0 ⊆ C1 ⊆ . . . Ck ⊆ Ck+1 ⊆ . . .,

if 〈nk〉k∈N is a cofinal subset of N, then⋃

k∈NCnk=⋃

n∈NCn and the equality also holds for their clo-sures. Hence:

lim infn

Cn =⋂

Σ∈N#∞

cl⋃k∈Σ

Ck = cl⋃k∈N

Ck (36)

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

K-limn

Cn = cl⋃k∈N

Ck (37)

What is the same, a decreasingly nested sequenceof sets is convergent in the sense of Painlene-Kuratowski and the result is stated in the followingproposition:

Proposition 19: Let 〈Cn〉n∈N be a decreasing se-quence of sets, i.e. C1 ⊇ C2 ⊇ . . .. Then, the limitlimnCn exists and is given by:

K-limn

Cn =∞⋂n=1

clCn (38)

The Painleve-Kuratowski convergence can be de-scribed using set inclusions which make it easyto check whether the K-lim of a given seuenceof sets exists. Since for any sequence of setsit is lim infnCn ⊆ lim supnCn then it will beK-limnCn = C whenever:

lim supn

Cn ⊆ C ⊆ lim infn

Cn (39)

It is therefore expedient to study under what condi-tions a given set is inside lim infnCn or is a supersetof lim supnCn. Some first results are given in thefollowing theorem:

Theorem 20: Let 〈Cn〉n∈N be a sequence of setsin a Hausdorff topological space (X , τ) and C be aclosed set. Then it is C ⊆ lim infnCn if and only if

for every V ∈ τ such that V ∩ C 6= ∅, there existsa N ∈ N∞ such that Cn ∩ V 6= ∅ for all n ∈ N .

The following theorem provides conditions forthe inclusion C ⊇ lim infnCn to hold.

Theorem 21: C ⊇ lim supnCn if and only if forevery compact set B ⊂⊂ X with B ∩C = ∅, thereexists N ∈ N∞ so that Cn ∩B = ∅ for all n ∈ N .

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

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

Theorem 22: Let (X , τ) be a Hausdorff topo-logical space and 〈Cn〉n∈N a sequence of subsetsof X . Let O ∈ τ . If whenever the set N ={n|Cn ∩O 6= ∅} is infinite, it is cofinite, then〈Cn〉n∈N is K-convergent.

REFERENCES

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

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

[3] G. Beer, On Convergence of Closed Sets in a Metric Space andDistance Functions, Bulletic of the Australian MathematicalSociety, vol. 31, 1985, pp. 421-432.