This document defines key concepts related to set convergence, including: 1) The inner and outer limits of a sequence of sets in topological and normed spaces, which describe the limit inferior and limit superior of the sets. 2) Properties of set convergence like the limit inferior and limit superior being characterized as intersections and unions of the sets over cofinal subsets of the natural numbers. 3) Characterizations of the inner and outer limits of sets in terms of open neighborhoods in topological spaces and open balls in normed spaces. 4) The inner limit of a sequence of sets in a normed space being the points for which the distance to the sets goes to zero as the index increases.

1. 1
On Set Convergence
Pantelis Sopasakis
Abstract—We give the definitions of inner and outer
limits for sequences of sets in tolopological 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
Painlev´ -Kuratowski convergence of set sequences.
e
Index Terms—Set Convergence, Inner limit, Outer
Limit, Limit inferior, Limit Superior.
I. I NTRODUCTION
finite, so it cannote be Σ ⊆ Nn , therefore there is a
σ ∈ Σ such that n ≤ σ. Thus, Σ is cofinal.
(2). Assume that Σ is a cofinal subset of N. Let
us assume that Σ is finite. Then, Σ has a maximal
element, let N . For every σ ∈ Σ, N + 1 σ. Hence
Σ is not cofinal. This contradicts our assumption,
therefore Σ is infinite.
Definition 4: Let Λ be any set. A set Φ ⊆ Λ is
called a cofinite subset of Λ if Φc = Λ Φ is finite.
Hereinafter, we shall denote the class of cofinite
subsets of N by N∞ .
Definition 5 (Inner Limit): Let Cn n∈N be a sequence of sets in a Hausdorff topological space
(X , τ ). The inner limit of this sequence is defined
as:
N what follows we always consider X to be a
set endowed with a Hausdorff topology τ which
we will denote by (X , τ ). The topology of the space
defines the class of open neighborhoods of points in
X:
∃N ∈ N∞ , ∃xv ∈ Cv
lim inf Cn = x
(2)
Definition 1: Let (X , τ ) be a topological space
v ∈ N, xv → x
n
and x ∈ X . The set of openneighborhoods of x is
Where the convergence xv → x is meant with
defined as:
respect to the topology τ .
(x) := {V ∈ τ |x ∈ V }
(1)
Accordingly, the outer limit of a sequence of sets
The topology of X governs the convergence of is defined as:
Definition 6 (Outer Limit): Let Cn n∈N be a sesequences of elements in X .
Definition 2: A sequence xn n∈N ⊆ X is said quence of sets in a Hausdorff topological space
to converge to some x ∈ X with respect to the (X , τ ). The outer limit of this sequence is defined
topology τ if for everh V ∈ (x) there is a N0 ∈ N as:
I
such that xk ∈ V for all k ≥ N0 .
We now introduce the notions of cofinal and
cofinite 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 cofinal
subset of Λ if for all λ ∈ Λ there exists a σ ∈ Σ
such that λ ≤ σ. We denote the cofinal subsets of
#
N by N∞ .
Proposition 1:
#
N∞ := {N ⊆ N, N is infinite}
Proof: (1). Let Σ be an infinite subset of N and
n ∈ N arbitrary. The set Nn = {m ∈ N, m ≤ n} is
Pantelis Sopasakis is with the National Technical University of
Athens, 9 Heroon Polytechneiou Street, 15780 Zografou Campus,
Athens, Greece. Email: chvng@mail.ntua.gr.
lim sup Cn =
n
x
#
∃N ∈ N∞ , ∃xv ∈ Cv
v ∈ N, xv → x
(3)
Where the convergence xv → x is meant with
respect to the topology τ .
If 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: The point-to-set distance on X is a
mapping d : X × 2X → [0, +∞] defined as
d (x, C) := inf { x − y ; y ∈ C}
y
(4)
The limit inferior and the limit superior of a sequence of real numbers will be of high importance
in what follows:

2. 2
Definition 8 (Inferior & Superior limit): The liProposition 2: The limit inferior of a sequence
mit inferior of a sequence an n∈N ⊆ R is defined of sets is:
∞ ∞
as:
D-liminf Cn =
Cm
(13)
lim inf a = lim inf a
(5)
n
n
n→∞
k≥n
n→∞
k
Accordingly, the limit superior of an
n∈N
lim sup an = lim sup ak
n→∞
n
n=1 m=n
Proof: First, let us define Bk = j≥n Cj . Then
the right hand side of equation (13) is written as
is:
(6)
Bn
k≥n
n∈N
The limit inferior of a sequence of elements or
(1). Assume that x ∈ Ci for all but finitely many
subsets of a space X (which does not need to be indices i. Then, there is a M ∈ N so that for all
endowed with any topology)
m ≥ M it is x ∈ Cm . We notice that if k ≥ M then
Definition 9: Let X be a set and An n∈N be a x ∈ C for all j ≥ k. Therefore,
j
sequence of sets in X . The limit inferior of An n∈N
is defined to be the set:
x∈
C =B
j
∃N ∈ N∞ , xv ∈ Cv
x
∀v ∈ N
D-liminf Cn =
k
j≥k
(7)
If on the other hand k ≤ M , then we can find
ˆ := max {k, M } so that
What is the same, we may define the limit supe- k
rior of a sequence of sets as:
Cj = Bk
x∈
ˆ
Definition 10: Let X be a set and An n∈N be a
ˆ
j≥k
sequence of sets in X . The limit superior of An n∈N
is defined to be the set:
Thus, for arbitrary index k ∈ N, there is always
ˆ ∈ N such that x ∈ Bˆ which means that x ∈
a k
#
k
∃N ∈ N∞ , xv ∈ Cv
D-limsup Cn = x
(8)
Bn .
n∈N
∀v ∈ N
n→∞
(2). Let us assume that
The limit inferior and the limit superior are ex∞
actly the inner and the outer limits when the space
Bn
x∈
X is endowed with the discrete topology, i.e. the
n=1
topology of the power set of X , τ = 2X . In all other
but there are infinitely many indices i, such that
cases, the inner and outer limit yield quite different
/
results that the limits inferior and superior. In all x ∈ Ci . Let sj j∈N ⊆ N be a strictly increasing
sequence such that x ∈ Csj - note that sj ≥ j. For
/
cases it holds:
any k ∈ N we have
n→∞
D-liminf Cn ⊆ lim inf Cn
n→∞
n→∞
(9)
x ∈ C sk ⊇
Consider for example the case of R with the usual
topology and the sequence of sets:
Cn =
Q, n is odd
R Q, n is even
Then,
D-liminf Cn = ∅
n→∞
C sj ⊇
sj ≥sk
which means that x ∈ Bk . This holds true for all
/
k ∈ N, thus x ∈ ∞ Bn which contradicts our
/ n=1
(10) 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
(11)
of sets is:
∞
while
lim inf Cn = Rn
n
Cj = Bk
j≥k
(12)
A well known property of D-liminf is stated as
follows:
∞
D-limsup Cn =
n→∞
Cm
(14)
n=1 m=n
Proof: The proof is analogous to the one of the
previous proposition and is omitted.

3. 3
we denote the unit ball of X by B := B (1). Then
II. T OPOLOGICAL C HARACTERIZATION
Proposition 4: Let Cn n∈N be a sequence of sets B (ε) = εB. This leads us to the following corollary:
Corollary 6: Let Cn n∈N be a sequence of sets
in a Hausdorff topological space (X , τ ). Then,
in a normed space (X , · ). Then,
lim inf Cn =
n
x
∀V ∈ (x) , ∃N ∈ N∞
∀n ∈ N : Cn ∩ V = ∅
(15)
lim inf Cn =
n
x|∀ε > 0, ∃N ∈ N∞ ,
∀n ∈ N : x ∈ Cn + εB
(19)
#
x|∀ε > 0, ∃N ∈ N∞ ,
∀n ∈ N : x ∈ Cn + εB
(20)
and
or equivalenty:
lim inf Cn =
lim sup Cn =
n
n
∀V ∈ (x) , ∃N0 ∈ N,
∀n ≥ N0 : Cn ∩ V = ∅
This corollary yields one equivalent characterization for the inner limit. The predicate “∀ε > 0”
Proof: (1). If x ∈ lim inf n Cn then we can translates into the intersection over all ε > 0,
#
find a sequence xk k∈N such that xk → x while the requirement “∃N ∈ N∞ ” will be written as
xk ∈ Cnk and nk k∈N ⊆ N is a strictly increasing the union over all N ∈ N∞ and then ∀n ∈ N
sequence of indices. For any V ∈ (x) there is a corresponds to an intersection. This allows us to
N0 ∈ N such that for all i ≥ N0 it is: xi ∈ V ; but restate the corollary using set elementary operations
also xi ∈ Cni . Thus Cni ∩ V = ∅. Therefore x is as follows:
Corollary 7: Let Cn n∈N be a sequence of sets
in the right-hand side set of equation (15).
(2). For the reverse direction assume that x be- in a normed space (X , · ). Then,
longs to the right-hand side set of equation (15).
Cv + εB
(21)
lim inf Cn =
Then, there is a strictly increasing sequence nk k∈N
n
ε>0 N ∈N∞ v∈N
such that for every V ∈
(x) we can find a
xk ∈ Cnk ∩ V . Hence, xk → x (in the topology and
τ ).
lim sup Cn =
Cv + εB
(22)
Or course for every result regarding the inner
n
# v∈N
ε>0 N ∈N
∞
limit, there is a corresponding one for its dual
object: the outer limit. Therefore, following similar
Before we state and prove the following result
steps with the ones in the proof of proposition 4 we we need to recall the following description of the
can show that:
closure of a set. Firstly, if (X , τ ) is a topological
Proposition 5: Let Cn n∈N be a sequence of sets space and C ⊆ X , its closure is defined as:
in a Hausdorff topological space (X , τ ). Then the
cl C :=
{F : F ⊇ C, F c ∈ τ }
(23)
outer limit of Cn n∈N is:
x
(16)
Then,
Proposition 8:
lim sup Cn =
n
#
∀V ∈ (x) , ∃N ∈ N∞ ,
∀n ∈ N : Cn ∩ V = ∅
x
(17)
or equivalenty:
lim sup Cn =
n

 ∀V ∈ (x) , ∃ nk k∈N ⊆ N 
nk k∈N ↑, ∀k ∈ N :
x
(18)


Cn k ∩ V = ∅
cl C = {x : ∀V ∈
(x) , V ∩ C = ∅}
(24)
Proposition 9: Let (X , τ ) be a Hausdorff topological space and Cn n∈N be a sequence of sets in
X . Then,
lim inf Cn =
n
cl
#
N ∈N∞
Cn
(25)
n∈N
Proof: (1). Let x ∈ lim inf n Cn and let Σ ∈
#
Instead of arbitrary open sets - if X is a normed N∞ . Let W be a neighborhood of x. There is a
space - we may use open balls, i.e. sets of the form: N0 ∈ N sucht that for all n ≥ N0 such that n ∈ Σ:
B (ε) = {x :
x < ε}
W ∩ Cn = ∅
(26)

4. 4
and take n2 ∈ Σ2 to the be smallest such that x2 ∈
(27) Cn2 . Eventually, we construct the aforementioned
sequences. We now have that:
Thus,
x ∈ cl
Cn
n∈Σ
(2). Assume that x ∈ lim inf n Cn . Then, there is
/
an open neighborhood of x, let W
x, such
that Σ0 := {n ∈ N|W ∩ Cn = ∅}. Therefore, x ∈
/
cl n∈Σ0 Cn . This completes the proof.
Proposition 9 reveals a very important property
of the inner limit:
Corollary 10: For any sequence of sets Cn n∈N
in a Hausdorff topological space (X , τ ) the limit
lim inf n Cn is closed.
Proof: The inner limit is given as an arbitrary intersection of closed set lim inf n Cn =
#
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,
lim inf Cn = x ∈ X | lim d (x, Cn ) = 0
n
n
(30)
Proof: (1). We firstly need to show that for any
x ∈ X it is
∞
∞
Cn ⊆ lim inf Cn
Cn ⊆ cl
n=1
1
1
, xk ∈ Cnk ⇒ 0 ≤ d (x, Cnk ) <
k
k
Therefore we have proven that limk d (x, Cnk ) = 0,
i.e. lim inf n d (x, Cn ) = 0.
The previous result is extended to provide a
particularly important description of the notion of
the inner limit on normed spaces. We use the pointto-set distance function to describe the inner limit
of a sequence of sets:
Proposition 13: Let (X , · ) be a normed space
and Cn n∈N be a sequence of sets in X . The inner
limit of a sequence of sets is:
xk − x <
n
n=1
(28)
lim sup d (x, Cn ) = 0
n
Let us assume that lim supn d (x, Cn ) > 0,
The following result is preliminary to what foli.e. there exists an increasing sequence of indices
lows. I put it here just because the proof has some
nk k∈N so that d (x, Cnk ) →k a > 0. This suggests
interesting steps. It is expedient to know beforehand
that there is a ε0 > 0 such that for all k ∈ N
that the interior limit of a sequence of sets in normed
one has that d (x, Cnk ) > ε0 . However, according
spaces is described by:
to proposition 9,
lim inf Cn = x ∈ X | lim d (x, Cn ) = 0
n
n
(29)
x ∈ cl
Cnk
k∈N
Proposition 12: Let (X , · ) be a normed space
and Cn n∈N be a sequence of sets in X . If x ∈
lim inf n Cn then lim inf n d (x, Cn ) = 0.
Proof: Let x ∈ lim inf n Cn . We will construct
a sequence xk k∈N ⊂ X and a strictly increasing
sequence of indices nk k∈N ⊆ N such that x −
1
xk ≤ k while xk ∈ Cnk . By proposition 9, if x ∈
#
lim inf n Cn , then for any Σ1 ∈ N∞ :
while
d x, cl
Cnk
≥ ε0
k∈N
which contradicts our initial assumption. Hence,
lim supn d (x, Cn ) = 0, i.e. limn d (x, Cn ) = 0 and
this way we have proven that x is in the right-hand
side set.
(2). Conversely, assume that x in the right-hand
x ∈ cl
Ci ⇒ ∃x1 ∈
C i : x − x1 < 1
side of (30), that is limn d (x, Cn ) = 0. For any
ε
i∈Σ1
i∈Σ1
ε > 0, we can find n0 ∈ N such that d (x, Cn ) ≤ 2
Choose n1 ∈ Σ1 to be the smallest integer such for all n ≥ n0 . By definition, we have that
that x ∈ Cn1 . Set Σ2 = {σ ∈ Σ1 , σ > n1 }. Note:
d (x, Cn ) = inf { x − y , y ∈ Cn }
Σ2 ⊆ Σ1 is cofinal for Σ1 . Then,
thus we can find a yn ∈ Cn such that
1
x ∈ cl
Ci ⇒ ∃x2 ∈
C i : x − x2 <
ε
yn − x < d (x, Cn ) + = ε
2
i∈Σ2
i∈Σ2
2

5. 5
That is:
∃ yn ∈ Cn :
yn − x < ε
Therefore, x ∈ Cn + εB from which it follows that
x ∈ lim inf n Cn (see corollary 6).
Note: For a sequence an n∈N , in order to show
that lim inf n an = a, it suffices to find a subsequence
of it that converges to a, i.e. it suffices to determine
a strictly increasing sequence nk k∈N ⊆ N such that
ank →k a, i.e. limk ank = a. Using this fact we can
prove the following:
Proposition 14: Let (X , · ) be a normed space
and Cn n∈N be a sequence of sets in X . The outer
limit of a sequence of sets is:
where K = cl B (x, ρ).
(1). First let us note that the inner limit in this
case is written as:
lim inf B (xn , ρn ) =
n
x
∀ε > 0, ∃N ∈ N∞ ,
∀k ∈ N, z ∈ B (xk , ρk + ε)
Assume that z ∈ cl B (x, ρ), or what is the same
that z − x ≤ ρ. For every ε > 0 there is a N =
ε
N (ε) ∈ N such that xk − x < 2 for all k ≥ N .
This means that for all k ≥ N ,
ε
z − xk ≤ z − x + xk − x < ρ +
2
But also ρk → ρ hence there is a M = M (ε) ∈ N
ε
lim sup Cn = x ∈ X | lim inf d (x, Cn ) = 0
such that for all k ≥ M , it is |ρk − ρ| < 2 . So, for
n
n
(31) k ≥ max {N, M } we have:
´
III. PAINLEN E -K URATOWSKI C ONVERGENCE
A sequence of sets Cn n∈N is said to converge
in the Painlen´ -Kuratowski sense if lim inf n Cn =
e
lim supn Cn . This common limit, whenever it exists,
will be denoted by K-limn Cn or simply limn Cn .
Since we already know that for any sequence
Cn c∈N it is lim inf n Cn ⊆ lim supn Cn , in order
to show that K-limn Cn exists and equals some set
C, it suffices to show that:
z − xk < ρk + ε ⇔ z ∈ B (xk , ρk + ε)
Thus, z ∈ lim inf n→∞ B (xn , ρn ). This way we have
proved that cl B (x, ρ) ⊆ lim inf n B (xn , ρn ).
(2). The second step is to prove that the outer
limit also converges to cl B (x, ρ). For that we use
the fact that
lim sup B (xn , ρn ) =
n
x
#
∀ε > 0, ∃N ∈ N∞ ,
∀k ∈ N, z ∈ B (xk , ρk + ε)
lim sup Cn ⊆ C ⊆ lim inf Cn
(32) This suggests that if z ∈ lim sup B (x , ρ ) then for
n n
n
all ε > 0, there exists a strictly increasing sequence
As it follows from the closedness properties of
of integers nk k∈N such that z ∈ B (xnk , ρk + ε)
the inner and the outer limit, the Kuratowski limit
for all k ∈ N. It takes similar actions as in (1) to
(whenever it exists) is a closed set. Later we will
complete the proof.
give conditions under which a closed set C satisfies
Under the same assumptions, the limit inferior of
an inclusion as in (32).
this sequence of open balls converges to an open
As a first example of a K-convergent sequence of
ball. For example it is:
sets we prove the following:
Proposition 15: Let xn n∈N be a sequence in Rn
D-liminf B (xn , ρn ) = B (x, ρ)
such that xn → x and and ρn n∈N ⊆ [0, ∞) with
n→∞
ρn → ρ < ∞. Then ,
A sequence of balls whose radii diverge to ∞
e
K-lim B (xn , ρn ) = cl B (x, ρ)
(33) converges (in the Painlev´ -Kuratowski sense) to the
n
whole space as stated in the following proposition:
Proposition 16: Let xn n∈N be a sequence in Rp
Proof: It would be a waste of time to calculate
the inner limit and the outer limit separately and such that xn → x and and ρn n∈N ⊆ [0, ∞) with
then corroborate that the both equal the closed ρn → ∞. Then ,
ball cl B (x, ρ). Instead, it suffices to show that the
K-lim B (xn , ρn ) = Rp
(34)
n
following inclusions hold:
and
(2)
(1)
lim sup B (xn , ρn ) ⊆ K ⊆ lim inf B (xn , ρn )
K-lim B (xn , ρn )c = ∅
(35)
n
n
n
n
n

6. 6
for every V ∈ τ such that V ∩ C = ∅, there exists
a N ∈ N∞ such that Cn ∩ V = ∅ for all n ∈ N .
The following theorem provides conditions for
the inclusion C ⊇ lim inf n Cn to hold.
Theorem 21: C ⊇ lim supn Cn if and only if for
every compact set B ⊂⊂ X with B ∩ C = ∅, there
exists N ∈ N∞ so that Cn ∩ B = ∅ for all n ∈ N .
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 subbasis), 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:
lim inf Cn =
cl
Ck = cl
Ck
(36)
Theorem 22: Let (X , τ ) be a Hausdorff topon
#
k∈Σ
k∈N
logical space and Cn n∈N a sequence of subsets
Σ∈N∞
Similarly we carry out the calculation for the of X . Let O ∈ τ . If whenever the set N =
{n|Cn ∩ O = ∅} is infinite, it is cofinite, then
lim sup from which it follows that
Cn n∈N is K-convergent.
K-lim Cn = cl
Ck
(37)
n
k∈N
R EFERENCES
A similar example refers to sequences of convex
polytopes.
Proposition 17: Let xi n∈N be a sequence of
n
n
points in a space (X , τ ) such that xi → xi for i ∈ I.
n
K
Then conv {xi }i → cl conv {xi }.
n
Limits of nested sequences of sets, either increasing or descreasing, are particularly easy to calculate.
Proposition 18: Let Cn n∈N be a nested and
increasing sequence of sets. Then it is convergent
in the Painlen´ -Kuratowski sense and K-limn Cn =
e
cl n∈N Cn .
Proof: Since C0 ⊆ C1 ⊆ . . . Ck ⊆ Ck+1 ⊆ . . .,
if nk k∈N is a cofinal subset of N, then k∈N Cnk =
n∈N Cn and the equality also holds for their closures. Hence:
What is the same, a decreasingly nested sequence
of sets is convergent in the sense of Painlen´ e
Kuratowski and the result is stated in the following
proposition:
Proposition 19: Let Cn n∈N be a decreasing sequence of sets, i.e. C1 ⊇ C2 ⊇ . . .. Then, the limit
limn Cn exists and is given by:
∞
K-lim Cn =
n
cl Cn
(38)
n=1
The Painlev´ -Kuratowski convergence can be dee
scribed using set inclusions which make it easy
to check whether the K-lim of a given seuence
of sets exists. Since for any sequence of sets
it is lim inf n Cn ⊆ lim supn Cn then it will be
K-limn Cn = C whenever:
lim sup Cn ⊆ C ⊆ lim inf Cn
n
n
(39)
It is therefore expedient to study under what conditions a given set is inside lim inf n Cn or is a superset
of lim supn Cn . Some first results are given in the
following theorem:
Theorem 20: Let Cn n∈N be a sequence of sets
in a Hausdorff topological space (X , τ ) and C be a
closed set. Then it is C ⊆ lim inf n Cn if and only if
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