Download BORNOLOGICAL CONVERGENCES A. Lechicki, S. Levi, and A

BORNOLOGICAL CONVERGENCES
A. Lechicki, S. Levi, and A. Spakowski
Dedicated to John Horvath, on the occasion of his eightieth birthday
Abstract. We study a family of convergences (actually pretopologies) in the hyperspace of a metric space that are generated by covers of the space. This family
includes the Attouch-Wets, Fell, and Hausdorff metric topologies as well as the lower
Vietoris topology. The unified approach leads to new developments and puts into
perspective some classical results.
Introduction
Let (X, d) be a metric space. For subsets C and D of X, the Hausdorff distance
between C and D is given by h(C, D) = inf{ε > 0 : C ⊆ B(D, ε) and D ⊆ B(C, ε)},
where B(A, ε) is the ε-enlargement of the set A of radius ε. The Hausdorff disH
tance induces a convergence H on the power set 2X by defining At → A whenever h(At , A) → 0. However, this convergence works well only when restricted to
bounded (closed) subsets. For unbounded sets convergence in the Hausdorff distance turns out to be too strong. There are simple examples of sequences of subsets
(such as the lines y = x/n in the plane) that do not converge with respect to the
Hausdorff distance, but should “reasonably” converge.
One solution to overcome this difficulty is to modify the convergence H by elements of a properly chosen family S ⊆ 2X . The most celebrated example of
this approach is the so-called Attouch-Wets convergence AW, also called boundedHausdorff convergence. This convergence was initially introduced by Mosco ([Mo])
and studied later by Attouch and Wets (see [AW1, ALW] and [AW2]). Fixing
x0 ∈ X, we say that a net (At ) of subsets AW-converges to A ⊆ X if for every
ε > 0 and n ∈ N,
At ∩ B(x0 , n) ⊆ B(A, ε) and A ∩ B(x0 , n) ⊆ B(At , ε), eventually.
In this case convergence in the Hausdorff distance has been modified by the family
S of bounded sets.
Attouch-Wets convergence has been intensively investigated since the mid 80’s
(see [Be] for references) and applied to study approximation and optimization problems. It turned out that besides the family of bounded subsets, other families were
also useful for modifying the convergence H. For instance, in the case of a Banach
2000 Mathematics Subject Classification. Primary 54B20; Secondary 54A20, 54E35.
Key words and phrases. Convergences, hyperspaces, covers, bornologies, pretopology, Hausdorff metric topology, uniform structure, totally bounded sets.
Typeset by AMS-TEX
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2
A. LECHICKI, S. LEVI, AND A. SPAKOWSKI
space X one can consider the family S of all norm (or weakly) compact subsets of
X ([BoVa]).
e
In this paper we present a general theory of S-convergences
which are modified
H-convergences through the use of various families S of subsets of X. It turns out
that all such convergences are pretopologies and that the families S are essentially
bornologies on X. This explains the title of the paper.
Besides the trivial bornology (S = 2X ) and the bornology of bounded sets, the
bornologies of finite sets, compact sets and totally bounded sets are also of general
interest: their common lower part is the lower Vietoris topology (Proposition 2.2.)
while the corresponding upper parts are, respectively, the cofinite topology, the
e
cocompact topology (so that the Fell topology is a particular S-convergence)
and
a newly identified object in the case of totally bounded sets.
The paper is organized as follows: in Section 1 we recall some definitions and fix
the notation. Section 2 is devoted to lower bornological convergences, called S − convergences. We study relationships between S − -convergences and other ”lower”
convergences, such as the lower Vietoris topology. Moreover, topological and uniform properties of S − are investigated. In Section 3 we discuss upper bornological
convergences S + . We show that there are distinct differences in the behaviour of
S + as compared to S − . And finally, in Section 4 we consider the convergence Se as
the supremum of S − and S + .
The authors would like to thank G. Beer for some very valuable discussions.
1. Preliminaries
Let Z be a non-empty set and let ϕZ denote the family of all filters on Z. For
each z ∈ Z, let Ul (z) denote the ultrafilter generated by {z}. A convergence on Z
is a mapping π from ϕZ to the family 2Z of all subsets of Z which satisfies the
following conditions:
(i) z ∈ π(Ul (z)) for all z in Z,
(ii) F ⊆ G implies π(F ) ⊆ π(G) for all F , G ∈ ϕZ.
The pair (Z, π) is called a convergence space. If F ∈ ϕZ and z ∈ π(F ), then we
say that F π-converges to z and we write z ∈ π-lim F . The notion of convergence
can be equivalently formulated in terms of nets (see e.g. [En, Gä]). Thus every
convergence π on Z can be treated as a mapping from the family of nets on Z into
2Z .
In this paper we will use both the net and the filter terminology.
If π1 and π2 are two convergences on Z, we say that that π2 is finer than π1 , or
that π1 is coarser than π2 , and write π1 ≤ π2 , provided z ∈ π2 -lim zt ⇒ z ∈ π1 lim zt for every net (zt ) on Z. Let πi , i T
∈ I, be a family of convergences on Z. For
each filter F on Z we denote: π(F ) = i∈I πi (F ). Then π is a convergence on Z
finer than
W each convergence πi , i ∈ I. We denote this uniquely defined convergence
with i∈I πi and call it the supremum of the convergences πi , i ∈ I.
Let (Z, π) be a convergence space and z ∈ Z. Let Nπ (z) be the filter obtained by
intersecting all filters that π-converge to z. This filter is called the π-neighbourhood
filter at z, and its elements are π-neighbourhoods of z.
A convergence π on Z is called pretopological (or a pretopology) if Nπ (z) πconverges to z for each z ∈ Z. If π is a pretopology on Z, the pair (Z, π) is called
a pretopological space.
BORNOLOGICAL CONVERGENCES
3
Let (X, U) be a quasi-uniform space and U−1 = {U −1 : U ∈ U} be the conjugate
quasi-uniformity, where U −1 = {(y, x) : (x, y) ∈ U }. If S is a nonempty family of
subsets of X, we may consider the following three convergences on 2X . Let (At ) be
a net of subsets of X and A ⊆ X. We say that the net (At ):
S−
S − -converges to A, and we write A ∈ S − -lim At or At → A, provided for each
S ∈ S and U ∈ U there exists t0 such that A ∩ S ⊆ U −1 (At ) for every t ≥ t0 ;
S+
S + -converges to A, and we write A ∈ S + -lim At or At → A, provided for each
S ∈ S and U ∈ U there exists t0 such that At ∩ S ⊆ U (A) for every t ≥ t0 ;
e
S-converges
to A provided it S − -converges to A and S + -converges to A,
where U (A) = {y ∈ X : (x, y) ∈ U for some x ∈ A}, U −1 (A) = {y ∈ X : (x, y) ∈
U −1 for some x ∈ A}.
For similar ideas see [BeLu, Pe, BaWe, BoVa, Sp].
In this paper the underlying space X is a metric space. We use the natural
uniformity of X, and the definitions of S − , S + and Se can be reformulated as
follows:
S−
At → A ⇔ ∀ S ∈ S ∀ ε > 0 ∃ t0 ∀ t ≥ t0 A ∩ S ⊆ Aεt ,
S+
A t → A ⇔ ∀ S ∈ S ∀ ε > 0 ∃ t 0 ∀ t ≥ t0 A t ∩ S ⊆ A ε ,
and,
Se
At → A ⇔ ∀ S ∈ S ∀ ε > 0 ∃ t0 ∀ t ≥ t0 A ∩ S ⊆ Aεt and At ∩ S ⊆ Aε ,
S
where Aε = B(A, ε) = x∈A B(x, ε) and B(x, ε) is the open ball with center x and
radius ε.
e
Note that if X is a metric space and S = {X} then the S-convergence
on the
space of closed bounded subsets of X is simply the H-convergence, i.e. the convergence in the Hausdorff metric. We also denote: H− = {X}− , H+ = {X}+ , and
V− , the lower Hausdorff, the upper Hausdorff, and the lower Vietoris convergence,
respectively.
We now define some useful set-theoretical operations on families of subsets:
↓ S = {A ⊆ X : A ⊆ S for some S ∈ S},
Σ(S) = {S1 ∪ S2 ∪ . . . ∪ Sn : Si ∈ S for i = 1, 2, . . . , n; n ∈ N}.
and we have the following properties:
S ⊆↓ S, S ⊆ Σ(S), ↓ Σ(S) = Σ(↓ S),
↓ {X} = 2X , ↓ {∅} = {∅}.
By s(X), cl(X), b(X), tb(X) and c(X) we denote the families of all singletons
of X, closed subsets of X, bounded subsets of X, totally bounded subsets of X and
compact subsets of X, respectively.
Let S, W ⊆ 2X be nonempty. We say that S refines W, and write S ≤ W, if for
every S ∈ S there exists W ∈ W with S ⊆ W . Observe that S ≤ W if and only if
S ⊆↓ W, and as a consequence: S ≤ W and W ≤ S if and only if ↓ S =↓ W.
A bornology on a set X is a family S of subsets of X such that: (1) S is a cover of
X, (2) S is closed under subsets and (3) S is closed under finite unions (see [HN]).
The set convergences defined above are generated by families of subsets which
are essentially bornologies.
From now on we assume that X is a metric space and d its metric.
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A. LECHICKI, S. LEVI, AND A. SPAKOWSKI
2. S − -convergences
We begin with the following observations on S − -convergences:
(2.1)
∅ ∈ S − -lim At for every S and every net (At ).
(2.2)
If A ∈ S − -lim At and B ⊆ A, then B ∈ S − -lim At .
(2.3)
S − = (↓ S)− = (Σ(S))− = (Σ(↓ S))− = (↓ (Σ(S)))− .
(2.4)
If S ≤ W, then S − ≤ W − .
Proposition 2.1. Let S be a cover of X. Then
(i)
V− ≤ S − ≤ H− .
(ii)
V− = s(X)− and H− = {X}− .
Proof. Let At and A be elements of 2X and ε > 0.
H−
S−
(i) If At → A, it is clear that At → A, as for every S ∈ S, A ∩ S ⊆ A ⊆ Aεt
S−
eventually. Suppose that At → A and that B(x, ε) ∩ A 6= ∅; pick y ∈ B(x, ε) ∩ A,
S ∈ S with y ∈ S and choose r > 0 such that d(x, y) + r < ε; then there exists t0
such that A ∩ S ⊆ Art for all t ≥ t0 and for every such t we can choose at ∈ At with
d(y, at) < r. Thus d(x, at ) ≤ d(x, y)+d(y, at) < d(x, y)+r < ε and B(x, ε)∩At 6= ∅.
V−
This shows that At → A.
V−
(ii) By (i) we only need to prove that s(X)− ≤ V− . Let At → A and pick
x0 ∈ X and ε > 0. We can suppose that x0 ∈ A; then B(x0 , ε) ∩ A 6= ∅ and there
exists t0 such that B(x0 , ε) ∩ At 6= ∅ for all t ≥ t0 . This shows that x0 ∈ Aεt for all
t ≥ t0 . Also, it is obvious that H− = {X}− .
Remark. If S contains a nonempty set then the S − -convergence is admissible if
and only if S is a cover of X. Recall that a convergence on a hyperspace is called
admissible if the mapping x 7→ {x} is an embedding. It is not hard to show that the
mapping x 7→ {x} is S − -continuous for every family S and if S is a cover of X then
the inverse mapping {x} 7→ x is continuous as well. Consequently, S − is admissible
provided S is a cover of X. Observe that this fact follows also from Proposition
−
2.1 (i), because V− and H− are admissible. Now assume that for S =
6 {∅}, SS
is
admissible and suppose that S is not a cover of X, i.e. there exists x ∈ X \ S.
S
S−
Take any y from S and put xn = y for n ∈ N. Then {xn } → {x} but xn 6→ x, a
contradiction.
Proposition 2.2. V− = S − if and only if every S ∈ S is totally bounded.
V−
Proof. Suppose each S ∈ S is totally bounded and let At → A, S ∈ S and
ε > 0; then
Sn A ∩ S is totally− bounded and there exist x1 , ..., xn ∈ A such that
A ∩ S ⊆ 1 B(xi , ε/2). By V -convergence of (At ) to A
∃t0 ∀t ≥ t0 At ∩ B(xi , ε/2) 6= ∅ for i = 1, ..., n;
pick yi ∈ At ∩ B(xi , ε/2); then
A∩S ⊆
n
S
i=1
B(xi , ε/2) ⊆
n
S
i=1
B(yi , ε) ⊆ Aεt for all t ≥ t0 .
Conversely, suppose that there exists
Sn S ∈ S which is not totally bounded; then
there exists ε > 0 such that S 6⊆ 1 B(xi , ε) for all {x1 , ..., xn} ⊆ S. If t =
BORNOLOGICAL CONVERGENCES
{x1 , ..., xn} and s = {y1 , ..., ym} belong to
and define the net (At ) on 2X by
At =
n
S
S∞
1
5
X k , put t ≤ s if and only if t ⊆ s
B(xi , ε/2).
i=1
ε/2
Then (At ) is not S − -convergent to S, as S is not contained in any At , while
V−
At → S; to see this pick y ∈ S, σ > 0 and put t0 = {y}. Then At ∩ B(y, σ) 6= ∅ for
all t ≥ t0 .
Corollary 2.3. For every cover S:
(i)
(localization) If A ⊆ X is such that A ∩ S is totally bounded for each S ∈ S,
V−
S−
then At → A if and only if At → A;
(ii)
V−
S−
If A is totally bounded, then At → A if and only if At → A.
Now, given two covers S and W, what are the conditions for S and W to generate
the same convergence?
We know that if S refines W, then S − ≤ W − but the reverse implication is
not true in general: take X = R2 , W the cover of all singletons and S = W ∪
{the unit disc}. By Proposition 2.2, S − = W − = V− , while S (or Σ(S)) is not a
refinement of W.
We can, however, proceed in the spirit of Proposition 2.2 provided the right
generalization of total boundedness is introduced:
Definition 2.4. Let S be a cover of X; a subset A of X is totally bounded with
respect to S, or A is S-totally bounded, if
∀ε > 0 ∃ S1 , ..., Sk ∈ S A ⊆ (A ∩ S1 )ε ∪ ... ∪ (A ∩ Sk )ε .
We note the following properties of S-total boundedness:
(a)
Every subset of each S ∈ S is S-totally bounded.
(b)
If A1 and A2 are S-totally bounded, then A1 ∪ A2 is S-totally bounded.
(c)
If S is the cover of all singletons, S-total boundedness is the usual total
boundedness.
If the cover S refines the cover W, then S-totally bounded sets are also
W-totally bounded.
(d)
Theorem 2.5. Let S and W be covers of X. The following are equivalent:
(i)
For all A ⊆ X and all S ∈ S, A ∩ S is W-totally bounded.
(ii)
The family of S-totally bounded subsets of X is included in the family of
W-totally bounded subsets of X.
S− ≤ W −.
(iii)
Proof. (i) ⇒ (ii). Let B ⊂ X be S-totally bounded and pick ε > 0; there exist
S1 , ..., Sm ∈ S such that B ⊂ (B ∩ S1 )ε/2 ∪ ... ∪ (B ∩ Sm )ε/2 ; by (i) the sets B ∩
S1 , ..., B ∩ Sm are W-totally bounded and there exist W11 , ..., Wk11 , ..., W1m , ...Wkmm ∈
W such that for j = 1, 2, ..., m,
B ∩ Sj ⊆ (B ∩ Sj ∩ W1j )ε/2 ∪ ... ∪ (B ∩ Sj ∩ Wkjj )ε/2 ;
thus
6
A. LECHICKI, S. LEVI, AND A. SPAKOWSKI
B⊆
kj
m S
S
j=1 r=1
(B ∩ Sj ∩ Wrj )ε ⊆
kj
m S
S
j=1 r=1
(B ∩ Wrj )ε .
(ii) ⇒ (i) is obvious.
W−
(i) ⇒ (iii). Suppose At → A and let S ∈ S and ε > 0 be given. There exist
Sk
ε/2
W1 , ..., Wk ∈ W such that A∩S ⊆ 1 (A∩S∩Wj )ε/2 and t0 such that A∩Wj ⊆ At
for all t ≥ t0 and j = 1, ..., m; therefore
A∩S ⊆
k
S
(A ∩ S ∩ Wj )ε/2 ⊆
j=1
k
S
j=1
(A ∩ Wj )ε/2 ⊆ Aεt
S−
for all t ≥ t0 . Hence At → A.
(iii) ⇒ (i). Suppose there exists A ⊆ X such that T = A ∩ S Sis not W-totally
m
bounded for some S ∈ S; then there exists ε > 0 such that T 6⊆ 1 S
(T ∩ Wi )ε for
∞
all W1 , ..., Wm ∈ W; put t = {W1 , ..., Wm } ⊂ W and order the set 1 W n as in
the proof of Proposition 2.2. Put Bt = (T ∩ W1 ) ∪ ... ∪ (T ∩ Wm ); then it is clear
W−
S−
that Bt → T and that Bt 6→ T , as T ∩ S = T 6⊆ Btε .
Let us denote by S⋆ the family of all S-totally bounded subsets of X.
Corollary 2.6.
(i)
S − = W − if and only if S⋆ = W⋆ .
(ii)
There are S − -convergences that are not comparable: take for example, X =
R2 and S and W the covers consisting of horizontal and vertical lines, respectively.
(iii)
Given A ⊆ X, every subset of A is S-totally bounded if and only if S − =
(S ∪ {A})− .
(iv)
S − = H− if and only if every subset of X is S-totally bounded, as {X}⋆ =
2X .
(v)
(localization) If A ⊂ X is such that for every W ∈ W the set A ∩ W is
S−
W−
S-totally bounded, then At → A ⇒ At → A.
In the list, (a) - (d), of properties of S-total boundedness, we did not mention
subsets; it turns out that, in general, S⋆ is not closed under subsets (see Example
2.7 below).
Another natural generalization of total boundedness is the following: say that
a subset B of X is weakly S-totally bounded if ∀ε > 0 ∃ S1 , ..., Sk ∈ S such that
B ⊆ S1ε ∪ ... ∪ Skε . It is clear that if S is the cover of all singletons, weak S-total
boundedness and S-total boundedness agree. Denote by S ⋆ the collection of all
weakly S-totally bounded subsets of X. Then S ⋆ is closed under finite unions and
subsets, S⋆ ⊆ S ⋆ and (S ⋆ )⋆ = S ⋆ . The next example shows however, that S ⋆ is too
large for our purposes:
Example 2.7. Let X = R2 , S the cover of vertical lines. The stripe B = {(x, y) :
0 ≤ x ≤ 1} is S-totally bounded, but its subset A = {(x, y) ∈ B : 0 < x ≤ 1, y =
1/x} is not; by Corollary 2.6, (S ∪ {B})− -convergence is strictly stronger than
S − -convergence, while S ⋆ = (S ∪ {B})⋆ , as B is weakly S-totally bounded.
As an application of Theorem 2.5, we have the
BORNOLOGICAL CONVERGENCES
7
Proposition 2.8. The following are equivalent:
(i)
S⋆ is closed under subsets.
(ii)
S − = S⋆− .
(iii)
S⋆ = (S⋆ )⋆ .
We now turn our attention to the question of convergence and uniform properties
of S − . The first observation is that S − -convergence need not verify the iterated
limit condition. Consequently, it is not topological in general.
Example 2.9. Let X = R2 , S the cover of horizontal lines and put A = {(x, y) :
x ≤ 0, y = 0}; for each n ∈ N consider
Bn = {(x, y) : x ≤ 0, y =
1
}.
2nex
S−
Then Bn → A, as for each horizontal line l and ε > 0, either A ∩ l = ∅ or
A ∩ l = A ⊆ Bnε for all n > 1/(2ε).
Now, for each n and m in N, consider the m horizontal lines
ki = {(x, y) : y =
i
}
2nm
(1 ≤ i ≤ m)
S−
n
n
n
= Bn ∩ {k1 ∪ ... ∪ km }. Then Cm
’s do
and put Cm
→ Bn as m → ∞. Also the Cm
−
n
not S -converge to A as A is unbounded and each Cm is finite.
However, non-topological convergences are not unusual in the theory of hyperspaces. For instance, the well-known Kuratowski convergence is also not topological
in general.
Non-topological convergences build a wide spectrum: from very general (no constraints at all) to fairly specialised (such as pretopologies) which are close to ordinary topologies (see e.g. [Ch] for more details on general convergences).
By definition, S − -convergence is a modification of the lower Hausdorff convergence H− by the family S. It is well-known (see e.g. [FLL]) that the convergence
H− can be described in terms of quasi-uniformities. The family {H−
ε : ε > 0},
−
X
X
ε
where Hε = {(A, B) ∈ 2 × 2 : A ⊆ B }, is a base of a quasi-uniformity on 2X
compatible with H− .
Let us consider the family S− = {S− (S1 , ..., Sn; ε) : S1 , ..., Sn ∈ S and ε > 0},
where
S− (S1 , ..., Sn; ε) = {(A, B) ∈ 2X × 2X : A ∩ Si ⊆ B ε for i = 1, ..., n}.
One might presume that this family generates a kind of uniform structure on 2X
that is related to S − . However, since S − is not topological in general, this structure
can not be expected to be quasi-uniform (quasi-uniformities are always compatible
with topological convergences).
Observe that the family S− is a filter-base. Let S− denote the filter on 2X × 2X
generated by S− . Of course, each element of S− contains the diagonal of 2X × 2X .
Consequently, the family S− is a pretopological uniform structure on 2X . It can
be shown that in general the filter S− is neither symmetric (i.e. S− 6= (S− )−1 ) nor
composable (S− 6⊆ S− ◦ S− ). This means that S− is neither a uniformity nor a
quasi-uniformity in general.
Now we come to the question of how S− is related to S − -convergence. It is
known (see e.g. [Ke, LZ]) that every pretopological uniform structure induces
8
A. LECHICKI, S. LEVI, AND A. SPAKOWSKI
a pretopology on the underlying space. Thus if S− is compatible with S − , the
convergence S − is a pretopology on 2X .
Pretopologies are convergences which have convergent neighbourhood filters (see
[Ch]). Thus they can be described in terms of total systems of neighbourhoods.
However, the main difference with topologies is that pretopologies need not admit
neighbourhood filters generated by open sets (this property is equivalent to the
iterated limit condition).
The total system of neighbourhoods of the pretopology λ(S− ) induced by S− on
2X is equal to {S− (A) : A ⊆ X}, where S− (A) denotes the section filter of S− at
A, i.e. the filter on 2X generated by the sets
(N)
S− (S1 , .., Sn; ε)(A) = {B ⊂ X : A ∩ Si ⊆ B ε for i = 1, .., n},
where S1 , ..., Sn ∈ S and ε > 0.
Lemma 2.10. The pretopological uniform structure S− is compatible with the convergence S − , i.e. λ(S− ) = S − .
Proof. Let S1 , ..., Sn ∈ S and ε > 0 be given. Take a net (At ) of subsets of
X. It follows immediately from (N) that At ∈ S− (S1 , ..., Sn; ε)(A) if and only if
A ∩ Si ⊆ Aεt for i = 1, .., n. Consequently, λ(S− ) = S − .
From Lemma 2.10 we infer the following
Theorem 2.11. The convergence S − is a pretopology on 2X . For each A ⊆ X the
family
BS − (A) = {S− (S1 , .., Sn; ε)(A) : S1 , ..., Sn ∈ S and ε > 0}
is a local base of S − at A.
Corollary 2.12. The convergence S − is topological if and only if
(∆− ) ∀A ⊆ X ∀ U ∈ BS − (A) ∃ V ∈ BS − (A) ∀B ∈ V ∃ W ∈ BS − (B) W ⊆ U.
Proof. Condition (∆− ) is just the translation, in terms of neighbourhood bases, of
the iterated limit condition (see [KT], [Gä, p.153]). Thus the corollary follows from
the theorem.
The following corollary follows immediately from Lemma 2.10.
Corollary 2.13. If the pretopological uniform structure S− is quasi-uniform, the
convergence S − is topological.
The reverse implication is not true in general.
Example 2.14. Let X be the space of real numbers and S the family of all finite
subsets of X. Then S − is topological by Proposition 2.1. We show that S− 6⊆
S− ◦ S− . Take y1 , y2 ∈ X such that |y1 − y2 | > ε > 0 and put S = {y1 , y2 }.
Then S− (S; ε) ∈ S− . Let V = S− (S1 , ..., Sn; δ) ∈ S− be arbitrary. Since S and Si
(i = 1, ..., n) are finite, there are c1 , c2 ∈ X \ (S ∪S1 ∪... ∪Sn ) such that |c1 −y1 | < δ
and |c2 −y2 | < δ. Put A = S, B = {y1 } and C = {c1 , c2 }. Then A∩Si ⊆ {y1 , y2 } ⊆
C δ and C ∩ Si = ∅ ⊆ B δ . Thus (A, B) ∈ V ◦ V but (A, B) 6∈ S− (S; ε) because
A ∩ S = {y1 , y2 } 6⊆ B ε .
The following is a sufficient condition for S− to be quasi-uniform.
BORNOLOGICAL CONVERGENCES
9
Proposition 2.15. Assume that
(ε)
∀ S ∈ S ∃ ε > 0 and S ′ ∈ S such that S ε ⊆ S ′ .
Then S− is a quasi-uniformity. Consequently, S − is topological.
Proof. It is enough to show that S− ⊆ S− ◦S− . Take an arbitrary S− (S1 , ..., Sn; ε) ∈
S− . Consider ε1 , ..., εn > 0 and S1′ , ..., Sn′ ∈ S such that Siεi ⊆ Si′ (i = 1, ..., n)
and pick σ, 0 < σ < min(ε/2, ε1 , ..., εn). Then V = S− (S1′ , ..., Sn′ ; σ) ∈ S− . If
(A, B) ∈ V ◦ V then there is C ⊆ X such that A ∩ Si′ ⊆ C σ and C ∩ Si′ ⊆ B σ
for i = 1, ..., n. We have to show that A ∩ Si ⊆ B ε for i = 1, ..., n. If x belongs
to A ∩ Si then x ∈ A ∩ Siεi ⊆ A ∩ Si′ ⊆ C σ . Hence there is c from C such that
d(c, x) < σ. Consequently, c ∈ (A ∩ Si′ )σ ⊆ Siεi ⊆ Si′ and c ∈ C ∩ Si′ ⊆ B σ . Finally,
x ∈ (B σ )σ ⊆ B ε .
Under some additional assumptions on S, the condition (ε) is also necessary.
Proposition 2.16. Suppose that S is a cover and S = Σ(S). If S− is quasiuniform, then S has the property (ε).
Proof. We can assume that card(X) > 1. Now suppose that condition (ε) is not
satisfied. Then there is S0 ∈ S such that
for every ε > 0 and every S ′ ∈ S, S0ε 6⊆ S ′ .
Because S is a cover and S = Σ(S) we can assume that card(S0 ) > 1. Take two
different elements s and r from S0 and pick δ, 0 < δ < (1/2)d(s, r). We will show
that for every ε > 0 and S1 , ..., Sn ∈ S
S− (S1 , ..., Sn; ε) ◦ S− (S1 , ..., Sn; ε) 6⊆ S− (S0 ; δ).
Take arbitrary ε > 0 and S1 , ..., Sn ∈ S. Then S0σ 6⊆ S = S1 ∪ ... ∪ Sn for any σ,
0 < σ < min(δ, ε). Pick any x ∈ S0σ \S. Then there is s0 ∈ S0 such that d(s0 , x) < σ.
We can assume that d(s0 , s) > δ, otherwise we would take r instead of s. Put
A = {x, s0 }, B = {s} and C = {x}. Then we have A ∩ Si ⊆ {s0 } ⊆ {x}σ ⊆ C ε and
C ∩ Si = ∅ ⊆ B ε for i = 1, 2, ..., n. But (A, B) 6∈ S− (S0 ; δ) because s0 ∈ A ∩ S\ B δ .
Observe that if S is a bornology, condition (ε) means that S is closed with
respect to small enlargements:
(E)
For every S ∈ S there is ε > 0 such that S ε ∈ S.
Thus for bornologies we have the following
Proposition 2.17. The structure S− is quasi-uniform if and only if S is closed
with respect to small enlargements.
Example 2.18. Let x0 ∈ X and consider the cover S of balls centered at x0 .
S − -convergence is the lower Attouch-Wets convergence AW− (see [Be, p.81]); since
S verifies the condition in Proposition 2.15, AW− is topological. As in general,
S − = (↓ S)− , AW− is equal to b(X)− , where b(X) is the family of all bounded
subsets of X. By Proposition 2.2 AW− = V− if and only if each bounded subset
of X is totally bounded, and AW− = H− if and only if X is bounded, by Corollary
2.6.(iv).
Remark. All results of this section are valid for uniform spaces.
10
A. LECHICKI, S. LEVI, AND A. SPAKOWSKI
3. S + -convergences
As with S − -convergences, we begin with a few observations:
(3.1)
X ∈ S + -lim At for every S and every net (At ).
(3.2)
If A ∈ S + -lim At and B ⊇ A, then B ∈ S + -lim At .
(3.3)
S + = (↓ S)+ = (Σ(S))+ = (Σ(↓ S))+ = (↓ (Σ(S)))+ .
(3.4)
If S1 ≤ S2 , then S1+ ≤ S2+ .
Proposition 3.1. Let S be a cover of X:
(i)
s(X)+ ≤ S + ≤ {X}+ .
(ii)
{X}+ = (2X )+ = H+ .
(iii)
A ∈ s(X)+ -lim At if and only if the upper limit Lsι At of (At ) with respect
to the discrete topology of X is contained in A.
Proof. (i) and (ii)Tare
clear as S is a cover. (iii) Suppose that A ∈ s(X)+ -lim At
S
and x ∈ Lsι At = t r≥t Ar ; then ∀ε > 0 ∃t0 ∀t ≥ t0 At ∩ {x} ⊆ Aε ; there exists
r ≥ t0 with x ∈ Ar and therefore x ∈ Aε ; thus Lsι At ⊆ A.
Conversely, suppose that Lsι At ⊆ A and fix p ∈ X and ε > 0; if p 6∈ Lsι At , there
exists t such that for all r ≥ t, p 6∈ Ar and Ar ∩ {p} ⊆ Aε ; if p ∈ Lsι At , p ∈ A and
At ∩ {p} ⊆ Aε for every t.
Remark. Note that a net (xt ) of points of X is s(X)+ -convergent to x ∈ X if and
only if y 6= x implies y 6= xt eventually; thus the restriction of s(X)+ to X is the
cofinite topology and s(X)+ is not admissible in general. It turns out that the only
admissible S + -convergence is H+ . Of course, H+ is admissible. If card(X) = 1
then S + = H+ . Assume now that card(X) > 1, S + is admissible and suppose that
S does not contain the set X. By (3.3) we can assume that S = Σ(S). Thus we
can direct S upwardly by inclusion and pick xS ∈ X \ S for each S ∈ S. Then the
net ({xS }) is S + -convergent to every subset of X. Taking any x, y ∈ X we infer
from admissibility of S + that (xS ) converges to x and y, i.e. x = y. Consequently,
card(X) = 1, a contradiction.
We now determine when two covers generate the same convergence.
Theorem 3.2. Let S and W be covers of X. The following conditions are equivalent:
(i)
S ≤ Σ(W).
(ii)
S+ ≤ W+.
Proof. (i) ⇒ (ii) follows from (3.3) and (3.4).
(ii) ⇒ (i). Suppose S does not refine Σ(W); then there exists S ∈ S such that
S 6⊆ W1 ∪ ... ∪ W
for every t = (W1 , ..., Wn)
Snn for every finite union of elements of W;
′
′
pick xt ∈ S \ 1 Wi . If t = (W1 , ..., Wn) and v = (W1 , ..., Wm
), put t ≤ v if and
′
′
only if W1 ∪ ... ∪ Wn ⊆ W1 ∪ ... ∪ Wm ; it is then easy to see that the net (xt ) is
W + -convergent - but not S + -convergent - to the empty set.
We remark that the empty set is isolated for S + if and only if X ∈ S, that is if
S + = H+ .
BORNOLOGICAL CONVERGENCES
11
Corollary 3.3. Let S and W be covers of X.
(i)
S + = W + if and only if ↓ Σ(S) =↓ Σ(W).
(ii)
If S + ≤ W + , then S − ≤ W − .
(iii)
S + = s(X)+ if and only if every S ∈ S is finite.
(iv)
S + = H+ if and only if there exist S1 , ..., Sk ∈ S such that X = S1 ∪ ... ∪ Sk .
(v)
s(X)+ = H+ if and only if X is finite.
(vi)
There are non comparable S + -convergences.
Theorem 3.2 and Corollary 3.3.(i) are in marked contrast with Theorem 2.5 and
Corollary 2.6.(i).
Also the example given before Definition 2.4 shows that, in general, S − ≤ W −
does not imply S + ≤ W + .
We will show later (see Proposition 3.6) that S + -convergence is not topological
in general. However, conditions which ensure that S + -convergence is topological
are similar to those for S − -convergence.
Using the same notation as in Section 2, let us consider the pretopological
uniform structure S+ on 2X × 2X generated by the family {S+ (S1 , ..., Sn; ε) :
S1 , ..., Sn ∈ S and ε > 0}, where
S+ (S1 , ..., Sn; ε) = {(A, B) ∈ 2X × 2X : B ∩ Si ⊆ Aε for i = 1, ..., n}.
Reasoning as in Section 2 we can prove the following
Theorem 3.4. The pretopological uniform structure S+ is compatible with the convergence S + . Consequently, S + is a pretopology on 2X and for each A ⊆ X the
family
BS + (A) = {S+ (S1 , ..., Sn; ε)(A) : S1 , ..., Sn ∈ S and ε > 0}
is a local base of S + at A.
Since S + is a pretopology, it is topological if and only if it verifies the condition
(∆+ ) ∀A ⊆ X ∀ U ∈ BS + (A) ∃V ∈ BS + (A) ∀B ∈ V ∃W ∈ BS + (B) W ⊆ U.
The condition (ε) from Proposition 2.15 works also for the structure S+ .
Proposition 3.5. Suppose that S is a cover and S = Σ(S). Then S+ is a quasiuniformity if and only if
(ε)
∀S ∈ S ∃ ε > 0 and S ′ ∈ S such that S ε ⊆ S ′ .
Although there are many similarities between the S − and S + -convergences, there
are also important differences. One of them is the behaviour of the convergence
s(X)+ . In contrast with s(X)− , s(X)+ is not topological in general.
Proposition 3.6. The convergence s(X)+ is topological if and only if (X, d) is
discrete.
Proof. If (X, d) is discrete then s(X)+ is topological by Corollary 3.3 (iii) and
Proposition 3.5. The necessary condition follows from Theorem 3.7 below.
12
A. LECHICKI, S. LEVI, AND A. SPAKOWSKI
Theorem 3.7. Assume that S = Σ(S). If S + is topological then every non-dense
subset S ∈ S has the property
(ε′ )
∃ ε > 0 and S ′ ∈ S such that S ε ⊆ S ′ .
Proof. Suppose that there is S0 ∈ S such that S0 6= X and S0ε 6⊆ S for every ε > 0
and every S ∈ S. We shall show that condition (∆+ ) does not hold. Let us take
x0 ∈ X and ε0 > 0 such that {x0 }ε0 ∩ S0 = ∅. Now put A = {x0 } and consider
the neighbourhood U = {D ⊆ X : D ∩ S0 ⊆ Aε } ∈ BS + (A). Take an arbitrary
V ∈ BS + (A) of the form V = {E ⊆ X : E ∩ Si ⊆ Aσ for i = 1, ..., k}. Then
1/n
1/n
S0 6⊆ S = S1 ∪ ... ∪ Sk for every n ∈ N. Pick xn ∈ S0 \ S for every n ∈ N.
Then the set B = {xn : n ∈ N} belongs to V. Now let W = {C ⊆ X : C ∩ Si′ ⊆
B δ for i = 1, ..., m} ∈ BS + (B) be arbitrary. Take n ∈ N such that 1/n < δ and
y ∈ B 1/n ∩ S0 . Then C = {y} ∈ W but C 6∈ U because C ∩ S0 6⊆ Aε .
Corollary 3.8. Suppose that S = Σ(S) and S ≤ S ∩ cl(X).
(i)
S + is topological if and only if condition (ε) is verified.
(ii)
If S + is topological then S − is topological.
(iii)
(2X , S + ) is topological if and only if (2X , S − , S + ) is bitopological.
Observe that if S =↓ S then the inequality S ≤ S ∩ cl(X) is equivalent to the
property that S is closed with respect to the closure operator: S ∋ S 7→ S ∈ S.
Applying Corollary 3.8 we infer that if a bornology S is closed with respect to the
closure operator then S + is topological if and only if the structures S− and S+ are
quasi-uniformities.
Notice that the converse of Corollary 3.8.(ii) is not true in general (see e.g.
Proposition 3.6).
Recall that c(X) and tb(X) are the families of all compact and totally bounded
subsets of (X, d), respectively.
Corollary 3.9.
(i)
c(X)+ is topological if and only if X is locally compact.
(ii)
tb(X)+ is topological if and only if X is locally totally bounded.
Proof. (i) The sufficient condition follows from Proposition 3.5. Now assume that
c(X)+ is topological and take any x ∈ X. If card(X) > 1, then {x} is not dense
in X. Since {x} ∈ c(X), it follows from Theorem 3.7 that {x}ε is contained in a
compact set for some ε > 0.
(ii) The proof is analogous.
We now look at the relationship between S + -convergences and some other “upper” convergences”. Following Beer [Be, p.44] we will consider the miss topology
µS on 2X determined by S. This topology has as a subbase all sets of the form
{A ⊆ X : A ⊆ S c }, where S ∈ S.
We say that a subset A of X strongly misses S, and write A ≪ S, if for each
S ∈ S, A ∩ S = ∅ implies Aε ∩ S = ∅ for some ε > 0. For A ⊆ 2X we write A ≪ S
if A ≪ S for each A ∈ A.
We will write S =↓cl S if S contains all sets of the form S ∩ C, where S ∈ S and
C ∈ cl(X). Notice that S =↓ S implies S =↓cl S.
BORNOLOGICAL CONVERGENCES
13
Proposition 3.10.
(i)
If S =↓cl S then S + ≤ µS .
(ii)
A ∈ S + -lim At implies A ∈ µS -lim At if and only if A ≪ S.
Proof. (i) Suppose At → A with respect to µS and fix S ∈ S, ε > 0; consider
S1 = S ∩ (Aε )c ∈ S; then A ∩ S1 = ∅ and thus At ∩ S1 = ∅ eventually. This implies
that At ∩ S ⊆ Aε eventually.
(ii) ⇒. Suppose that there is S0 ∈ S such that A ∩ S0 = ∅ and Aε ∩ S0 6= ∅ for
each ε > 0. Put An = A1/n . Then A ∈ S + -lim An but (An ) does not µS -converge
to A.
(ii) ⇐. Suppose At → A with respect to S + and A ∩ S = ∅; then Aε ∩ S = ∅ for
some ε > 0 and At ∩ S ⊆ Aε eventually; thus At ∩ S = ∅ eventually.
Corollary 3.11.
(i)
If S =↓cl S then S + = µS on A if and only if A ≪ S.
(ii)
(iii)
(iv)
If S =↓cl S and A ≪ S then S + restricted to A is topological.
The convergences s(X)+ and c(X)+ restricted to cl(X) are topological.
The convergence tb(X)+ restricted to c(X) is topological.
It follows from Corollary 3.11.(i) that not every topological S + -convergence is
equal to the topology µS on cl(X). Indeed, b(X)+ is topological by Proposition
3.5 but cl(X) does not strongly miss b(X) in general.
From Corollary 3.11 we infer also that s(X)+ amounts to the co-finite and c(X)+
to the co-compact topology on cl(X). This leads to the
Corollary 3.12. (cf. [Be, Theorem 5.1.6]). If S is the family of all compact subsets
of X then the S + -convergence coincides with the co-compact topology on cl(X).
It is well-known (see e.g. [KT, p.43]) that H+ -convergence implies the upper Kuratowski convergence K+ on cl(X). So it is natural to ask which S + -convergences
imply the
K+ . Recall that At → A for K+ if Ls At ⊆ A, where
T convergence
S
Ls At = t s≥t As . We begin with the following
Proposition 3.13.
(i)
If a net (At ) is S + -convergent to A, then for every S ∈ S, Ls (At ∩ S) ⊆ A.
(ii)
Let S be the family of all compact subsets of X. Then a net (At ) is S + convergent to A if and only if for every S ∈ S, Ls (At ∩ S) ⊆ A.
Proof. (i) Let S ∈ S and suppose that x ∈ Ls (At ∩ S); fix ε > 0; there exists t0
such that for all t ≥ t0 , At ∩ S ⊆ Aε/2 , and for some t ≥ t0 , B(x, ε/2) ∩ At ∩ S 6= ∅;
thus x ∈ Aε for all ε > 0 and x ∈ A.
(ii) Apply (i), Proposition 3.10 and [Be, Proposition 5.2.5].
If follows from (ii) of the above proposition that K+ ≥ c(X)+ on cl(X).
The following lemma was pointed out to us by G. Beer:
Lemma 3.14. If S =↓ S then the following conditions are equivalent:
(i)
∀x ∈ X ∃ δ > 0 such that {x}δ ∈ S.
(ii)
For every net (At ) which is S + -convergent to A, we have Ls At ⊆ A.
14
A. LECHICKI, S. LEVI, AND A. SPAKOWSKI
Proof. (i) ⇒ (ii). Let x ∈ Ls At and take ε > 0. Pick 0 < δ < ε/2 such that
{x}δ ∈ S and proceed as in the proof of Proposition 3.13.(i).
(ii) ⇒ (i). Suppose there is x0 ∈ X such that {x0 }1/n 6∈ S for every n ∈ N.
Then for every n ∈ N and S ∈ S there is x(n,S) ∈ {x0 }1/n \ S. It is clear that the
net (x(n,S) ) is S + -convergent to the empty set, while x0 ∈ Ls {x(n,S) }.
As a counterpart to the preceding lemma, we have the
Lemma 3.15. If K+ is stronger than S + , every S ∈ S is relatively compact.
Proof. Suppose there exists an S ∈ S which is not relatively compact; let (xn ) be a
sequence in S without converging subsequence and put An = {xm : m > n}. Then
Ls An = ∅ but (An ) is not S + -convergent to the empty set as An ∩ S is not empty
for every n.
An application of the two previous lemmas yields:
Corollary 3.16. K+ = S + if and only if X is locally compact and S generates the
bornology of relatively compact subsets (equivalently, S + is the co-compact topology).
Let us now consider the bornology tb(X) of all totally bounded subsets of (X, d).
Of course, c(X)+ ≤ tb(X)+ .
Theorem 3.17. The following conditions are equivalent:
(i)
The metric d is complete.
(ii)
tb(X)+ = c(X)+ .
(iii)
tb(X)+ ≤ K+ on cl(X).
Proof. (i) ⇒ (ii). If d is complete then tb(X) ≤ c(X) and (3.4) implies (ii).
(ii) ⇒ (iii) is obvious.
(iii) ⇒ (i). By Lemma 3.15 each totally bounded subset of X is relatively
compact; hence d is complete.
By Lemma 3.14, K+ ≤ tb(X)+ on cl(X) if and only if (X, d) is locally totally
bounded. Thus we have
Corollary 3.18. K+ = tb(X)+ on cl(X) if and only if (X, d) is locally compact
and complete.
We end this section with a reference to b(X)+ , where b(X) is the family of all
bounded subsets of X. b(X)+ is topological by Proposition 3.5 and it is equal to
the upper Attouch-Wets convergence AW+ , as described in [Be, p.81]. By Corollary
3.3, AW+ = s(X)+ if and only if each bounded subset of X is finite and AW+ = H+
if and only if X is bounded.
The relationship of the main S + -convergences on cl(X) is expressed by the
following diagram:
BORNOLOGICAL CONVERGENCES
15
H+
AW +
K+
tb(X) +
c(X)+
s(X)+
Remark. All results of this section are valid for uniform-spaces.
e
4. S-convergences
We start with a result linking Se and S + convergences:
Lemma 4.1. A net (At ) is S + -convergent to A if and only if the net (At ∪ A) is
e
S-convergent
to A.
Proof. ⇒. If (At ) is S + -convergent to A, given S and ε > 0, (At ∪ A) ∩ S =
(At ∩ S) ∪ (A ∩ S) ⊆ Aε eventually; also it is clear that (At ∪ A) is S − -convergent
to A.
e
⇐. If (At ∪ A) is S-convergent
to A, (At ∪ A) is S + -convergent to A and
ε
(At ∪ A) ∩ S ⊆ A eventually, so that At ∩ S ⊆ Aε eventually.
Corollary 4.2. Let S and W be covers of X. Then
f if and only if S + ≤ W + .
(i)
Se ≤ W
(ii)
f if and only if S + = W + and this implies S − = W − .
Se = W
f
W
f and At S→ A then by Lemma 4.1 At ∪ A →
Proof. If Se ≤ W
A. Applying again
+
Lemma 4.1 we get At → A with respect to W . The necessary condition follows
from Corollary 3.3 (ii).
+
(ii) follows from (i).
e Thus Se
Remark. Note that the mapping x 7→ {x} is continuous with respect to S.
−
is admissible because it is finer than S which is admissible.
e For S1 , ..., Sn ∈ S and
Now we come to the question of uniform properties of S.
ε > 0 put
S(S1 , ..., Sn; ε) = {(A, B) ∈ 2X × 2X : A ∩ Si ⊆ B ε , B ∩ Si ⊆ Aε , i = 1, ..., n}.
Then the family {S(S1 , ..., Sn; ε) : S1 , ..., Sn ∈ S and ε > 0} is a base of the pree = S− ∨ S+ .
topological uniform structure S
From theorems 2.11 and 3.4 we infer
16
A. LECHICKI, S. LEVI, AND A. SPAKOWSKI
e is compatible with the conTheorem 4.3. The pretopological uniform structure S
e Consequently, Se is a pretopology on 2X and for each A ⊆ X the family
vergence S.
BSe (A) = {S(S1 , ..., Sn; ε)(A) : S1 , ..., Sn ∈ S and ε > 0}
is a local base of Se at A.
Of course, Se = S − ∨S + and Se is topological if and only if it verifies the condition
(∆)
∀A ⊆ X ∀ U ∈ BSe (A) ∃ V ∈ BSe (A) ∀B ∈ V ∃ W ∈ BSe (B) W ⊆ U.
Combining Propositions 2.15 and 3.5 we obtain.
e is a uniformity
Proposition 4.4. Suppose that S is a cover and S = Σ(S). Then S
if and only if
(ε)
∀S ∈ S ∃ ε > 0 and S ′ ∈ S such that S ε ⊆ S ′ .
Remark. It follows from Proposition 4.4 and Corollary 3.8 that if a bornology S is
e
closed with respect to the closure operator, then S + is topological if and only if S
is a uniformity.
e
We will now look at the separation properties of S-convergences.
Recall that a pretopological space (Z, π) is T1 if {z} is closed for each z ∈ Z,
T2 or Hausdorff if every two distinct elements of Z have disjoint neighbourhoods,
and, is T3 or regular if for each z ∈ Z and U ∈ Nπ (z) there is V ∈ Nπ (z) such that
V ⊆ U.
Te
is contained in the diagonal
Since Se is a pretopology, it is T1 if and only if S
X
X
X e
of 2 × 2 . Consequently, (2 , S) is not T1 in general. Indeed, take A ⊆ X such
e
that A 6= A. Then (A, A) belongs to every element of S.
Proposition 4.5.
e is T1 if and only if S is a cover of X.
(i)
(cl(X), S)
e is T2 if and only if the condition (i) in Lemma
(ii)
Let S =↓ S. Then (cl(X), S)
3.14 is verified.
Proof. (i) ⇒. Suppose that there is x ∈ X such that x 6∈ S for every S ∈ S. Then
e A contradiction.
({x}, ∅) belongs to every element of element of S.
⇐. An easy proof is omitted.
(ii) ⇒. Suppose there exists x ∈ X that does not verify the condition; then for
every n and every S ∈ S there exists x(n,S) ∈ {x}1/n \ S. It can be checked that
e
e
the net ({x(n,S) }) S-converges
to both {x} and ∅; therefore the S-convergence
is
not Hausdorff.
⇐. Let A and B be closed and A 6= B. We can assume that there is x ∈ A \ B.
Then {x}ε ∩ B = ∅ for some ε > 0. Take 0 < δ < ε/2 such that S = {x}δ ∈ S. Put
U = S− (S; δ)(A) and V = S+ (S; δ)(B). Then U ∩ V = ∅.
It follows from Proposition 4.5.(i) and Proposition 4.4 that if a bornology S
e is a Hausdorff uniformity on
is closed with respect to small enlargements then S
e is a completely regular topological space.
cl(X) × cl(X) and (cl(X), S)
BORNOLOGICAL CONVERGENCES
17
Proposition 4.6. Let S be a bornology. The following conditions are equivalent:
e
(i)
The empty set has a closed local base for S.
(ii)
For every S ∈ S, there exists S ′ ∈ S such that S ⊆ int (S ′ ).
(iii)
∀A ∈ cl(X) ∀S ∈ S ∀ε > 0 ∃ S ′ ∈ S S(S ′ ; ε/2)(A) ⊆ S+ (S; ε)(A).
Se
Proof. (i) implies (ii). Let S be in S; put S+ (S)(∅) = {B : B ∩ S = ∅}; then there
Se
exists S ′ ∈ S such that S(S ′ )(∅) ⊆ S+ (S)(∅); suppose that S 6⊂ int (S ′ ); then there
exists x in S which does not belong to int (S ′ ) and we can find a sequence (xn ) in
the complement of S ′ that converges to x. Thus, for every n, {xn } ∈ S+ (S ′ )(∅),
+
e
and {x} = S-lim{x
n } must be in S (S)(∅), a contradiction.
e
Note that condition (i) implies that the S-convergence
is Hausdorff.
(ii) implies (iii). Let S be in S and consider S ′ ∈ S given by (ii); take any
Se
e
D ∈ S(S ′ ; ε/2)(A) and let (Dt ) be a net of elements in S(S ′ ; ε/2)(A) which Sσ
′
converges to D; if x ∈ D ∩ S, there is 0 < σ < ε/2 such that {x} ⊆ S and
D ∩ {x}σ ⊆ Dtσ eventually; thus d(x, dt) < σ for some t and some dt ∈ Dt . Since
Dt ∈ S(S ′ ; ε/2)(A), we obtain that dt ∈ Aε/2 and finally that x ∈ Aε .
(iii) implies (i). Apply (iii) to the empty set.
Remarks.
(a) If no S ∈ S is dense, a weaker form of (iii) with A 6= ∅ implies (i).
(b) If S ≤ S ∩cl(X), (ii) is equivalent to ∀S ∈ S, ∃ S ′ ∈ S such that S ⊆ int (S ′ ).
Corollary 4.7. The small enlargement condition for S implies regularity for
e which, in turn, implies condition (ii) above.
(cl(X), S),
We close the paper with a word on hit-and-miss topologies: if ∆ is a subfamily
of cl(X), the hit-and-miss topology determined by ∆ is the supremum of the lower
Vietoris topology and of the miss topology µ∆ (see, for instance, [BT1], [BT2],
[HL], [Po] and [Be]). Under mild assumptions on the cover S, we have V− ≤ S −
e
and S + ≤ µS by Propositions 2.1.(i) and 3.10.(i). Thus the S-convergence,
which
is more ”symmetric” in nature, is not comparable, except for the Fell topology, to
the corresponding hit-and-miss topology.
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A. Lechicki, Softlab, Zamdorfer Str. 120, 81677 Munich, Germany,
[email protected]
S. Levi, University of Milano-Bicocca, Dipartimento Di Matematica E Applicazioni, Via Bicocca degli Arcimboldi 8, 20126 Milano, Italy, [email protected]
A. Spakowski, Opole University, Instytut Matematyki i Informatyki, Oleska 48,
45-052 Opole, Poland, [email protected]