@
Applied General Topology

c© Universidad Politécnica de Valencia

Volume 4, No. 2, 2003

pp. 445–465

Graph topologies on closed multifunctions

Giuseppe Di Maio, Enrico Meccariello and Somashekhar

Naimpally

Dedicated by the first two authors to Professor S. Naimpally on the occasion of
his 70th birthday.

Abstract. In this paper we study function space topologies on
closed multifunctions, i.e. closed relations on X ×Y using various hy-
pertopologies. The hypertopologies are in essence, graph topologies
i.e topologies on functions considered as graphs which are subsets of
X ×Y . We also study several topologies, including one that is derived
from the Attouch-Wets filter on the range. We state embedding theo-
rems which enable us to generalize and prove some recent results in the
literature with the use of known results in the hyperspace of the range
space and in the function space topologies of ordinary functions.

2000 AMS Classification: 54B20, 54C25, 54C35, 54C60, 54E05, 54E15.
Keywords: hyperspaces, function spaces, graph topologies, Vietoris topology,
Fell topology, uniform convergence on compacta, U-topology, ∆-topology, prox-
imal ∆-topology, ∆U-topology, proximal ∆U-topology, Hausdorff-Bourbaki
uniformity, ∆-Attouch-Wets filter.

1. Introduction.

Recently McCoy [24] studied relations among four hyperspace topologies
(viz. Fell topology, Fell uniform topology, Vietoris topology and Hausdorff-
Bourbaki topology) and the corresponding topologies on set valued maps. In
this paper we plan to study the subject comprehensively in more general situ-
ations. We recall that for a topological space Z the hyperspace, 2Z, of closed
subsets of Z has a number of natural topologies on it obtained from the topo-
logy on Z. In our setting (X,τ1) and (Y,τ2) denote Hausdorff topological spaces
and Z the product space X×Y equipped with the product topology τ = τ1×τ2.
If δ1 and δ2 are compatible proximities on X and Y respectively, then on Z is
assigned the product proximity δ = δ1 × δ2. The hyperspace 2Z = 2X×Y can
be considered as the space F of all set valued maps on X to 2Y taking points



446 G. Di Maio, E. Meccariello and S. Naimpally

of X to (possibly empty) closed subsets of Y . We do not distinguish between
a function f ∈ F and its graph {(x,f(x)) : x ∈ X} ⊂ Z = X ×Y . Thus our
study includes topologies on the spaces of partial maps studied first in 1936
and which are being studied intensively in recent times ([1], [2], [3], [13], [19],
[20], [23], [27], [33], [34]).

Given a Hausdorff topological space Z, for each subset E of Z, clZE, intE
and Ec stand for the closure, interior and complement of E in Z. Moreover

E− = {A ∈ 2Z : A∩E 6= ∅};
E+ = {A ∈ 2Z : A ⊂ E}.

Futhermore, if δ is a compatible proximity on Z (for details see [30]), we set

E++δ = {A ∈ 2
Z : A �δ E}. (Note: A �δ E iff A6δEc where 6δ denotes

the negation of δ).

We omit δ if it is clear from the context and write E++δ simply as E
++.

We recall that the set of all compatible proximities on Z is partially ordered
as follows: δ1 ≤ δ2 iff whenever A, B ⊂ Z and A6δ1B, then A6δ2B (see [30]).

Some special cases of δ are:

δ0 the fine LO-proxmity on Z given by
Aδ0B iff clZA∩ clZB 6= ∅.
δ0 is called the Wallman proximity.

It is well known that δ0 is, by far, the most important compatible LO-
proximity on Z, and that δ0 is EF iff Z is normal (Urysohn’s Lemma).

If Z is Tychonoff and V is a compatible uniformity on Z, then
δ(V) denotes the EF-proximity on Z given by
Aδ(V)B iff V (A) ∩B 6= ∅ for each V ∈V.
δ(V) is called the uniform proximity (induced by V).

If Z is a metrizable space with metric d, then

δ(d) is the EF-proximity on Z given by
Aδ(d)B iff Dd(A,B) = inf{d(a,b) : a ∈ A,b ∈ B} = 0.
δ(d) is called the metric proximity (induced by d).

For Z = X ×Y , we use the symbol ∆ (resp. ∆1, ∆2) to denote a subfamily
of CL(Z) = 2Z \{∅} (resp. of CL(X) = 2X \{∅}, CL(Y ) = 2Y \{∅}) which
is a cobase, i.e.

(a) is closed under finite unions; and
(b) contains the singletons.
A cover is a cobase which is also closed hereditary.
Moreover, we assume
(c) ∆1 × ∆2 ⊂ ∆.
In some cases, in addition to the above condition, we suppose



Graph topologies on closed multifunctions 447

(d) p1(∆) ⊂ ∆1 and p2(∆) ⊂ ∆2, where p1 and p2 are projections from Z
to X and Y respectively.

A typical and important example of a cover is ∆ = K(Z), the
family of all nonempty compact subsets of Z, ∆1 = K(X), ∆2 = K(Y ).
Moreover, in this case (c) − (d) also hold.

In what follows, unless explicitly stated, we assume always that ∆ ⊂ CL(Z)
(resp. ∆1 ⊂ CL(X), ∆2 ⊂ CL(Y )) is a cobase.

We now describe some hypertopologies on 2Z (for details see [4]). Suppose
δ is a compatible LO-proximity on Z.

The lower Vietoris topology τ−V on 2
Z has a subbase {W− : W ∈ τ}.

The upper ∆-topology τ(∆)+ on 2Z has a base {W + : Wc ∈ ∆}.
The upper proximal ∆-topology σ(∆,δ)+ on 2Z has a base {W ++ : Wc ∈ ∆}.
The ∆ topology τ(∆) on 2Z equals τ−V ∨ τ(∆)

+.
The proximal ∆ topology σ(∆,δ) on 2Z equals τ−V ∨σ(∆,δ)

+.
The upper ∆U-topology τ(∆U)+ on 2Z has a base
{W + : Wc ∈ ∆ or clW ∈ ∆}.
The upper proximal ∆U topology σ(∆U,δ)+ on 2Z has a base
{W ++ : Wc ∈ ∆ or clW ∈ ∆}.
The ∆U-topology τ(∆U) on 2Z equals τ−V ∨ τ(∆U)

+ ([8] and [16]).
The proximal ∆U-topology σ(∆U,δ) on 2Z equals τ−V ∨σ(∆U,δ)

+.

Special cases:

(1) Vietoris and proximal topologies: when ∆ = CL(Z),

the upper Vietoris topology τ(V )+ = τ(CL(Z))+;
the Vietoris topology τ(V ) = τ(CL(Z));
the upper proximal topology σ(δ)+ = σ(CL(Z),δ)+ and
the proximal topology σ(δ) = σ(CL(Z),δ) = σ, if δ is understood.

The paper [7] deals with only metric proximities, and [18] remains
unpublished. It is not widely known that proximal hypertopologies
can be studied in more general situations and not merely in metric
spaces. (However, see the recent papers [11], [12] and [16]).

We note that the Vietoris topology is itself a proximal topology, i.e. τV =
σ(δ0).

(2) Fell topology: when ∆ = K(Z),

the upper Fell topology (also called the co-compact topology) τ(F)+ = τ(K(Z))+;
the Fell topology τ(F) = τ(K(Z));



448 G. Di Maio, E. Meccariello and S. Naimpally

the U-topology τ(U) = τ(K(Z)U) (see [8]).

When ∆ = K(Z) and δ is EF, we have
τ(F) = τ(K(Z)) = σ(K(Z),δ) = σ(F,δ) and
τ(U) = τ(K(Z)U) = σ(K(Z)U,δ) = σ(U,δ).

In this case the Fell topology equals the proximal Fell topology and
this explains the reason for several beautiful results. In generalizing
results concerning Fell topology to ∆-topologies, we find that some are true in
τ(∆) while others are true in σ(∆)! (Also see below about weak topologies).

(3) Ball and proximal ball topologies:

When Z is a metric space, ∆ = B is the cobase generated by all finite unions
of all closed balls of nonnegative radii and δ is the metric proximity, we have

the ball topology τ(B) = τ(∆) ([4]);
the proximal ball topology σ(B) = σ(∆,δ) ([17]).

The proximal ball topology is very close to the Wjisman topology. In fact,
the two are equal in metric spaces satisfying some simple conditions that are
present in a normed linear space ([17]).

For Z = X×Y , when we wish to refer to hypertopologies on 2Y , we use the
suffix 2 e.g.
τ2(V ) denotes the Vietoris topology on 2Y ;
τ2(F) denotes the Fell topology on 2Y ;
σ2(δ2) = σ2 denotes the proximal topology w.r.t. δ2 on 2Y etc..

(4) Weak topologies:

If Z = X×Y , then for each of the topologies involving ∆ described above, we
also have an associated weak topology wherein ∆ is replaced by ∆1×∆2 (see
[31]) and we attach the letter ”w”. Thus τ(w∆) = τ(∆1 × ∆2) and important
special examples are:

the weak Vietoris topology τ(wV ) = τ(CL(X) × CL(Y )) ⊂ τ(V ) =
τ(CL(Z));

the weak Fell topology τ(wF) = τ(K(X) ×K(Y )) ⊂ τ(F) = τ(K(Z)).

For single-valued functions with closed graphs, it was shown in [21] that
τ(wF) = τ(F). The proof also works for F and combining this result with the
fact that when the proximity δ is EF, τ(F) = σ(F,δ) we have
τ(wF) = τ(F) = σ(wF,δ) = σ(F,δ).



Graph topologies on closed multifunctions 449

In generalizing McCoy’s results involving Fell topology we find
that our generalizations hold if we replace an appropriate member
from the above four.

(5) Hausdorff-Bourbaki and Attouch-Wets topologies:

Definition 1.1. Let Y be a Tychonoff space, V a compatible uniformity and
∆2 ⊂ CL(Y ).

(i) For each V ∈V set:
VH = {(A,B) ∈ 2Y × 2Y : A ⊂ V (B) and B ⊂ V (A)}.
The family {VH : V ∈V} is a base for a uniformity VH on 2Y called

the Hausdorff-Bourbaki uniformity (cf. [4]) (or the HB-uniformity for
short).

(ii) Whereas for each D ∈ ∆2 and V ∈V set:
[D,V ] = {(A,B) ∈2Y ×2Y : A∩D ⊂V (B) and B ∩D ⊂V (A)}.
The family {[D,V ] : D ∈ ∆2 and V ∈ V} is a base for a filter V∆2

on 2Y called the ∆2-Attouch-Wets filter (cf. [5], [6] and [16]) (or the
∆2-AW filter for short).

Remark 1.2. Let Y be a locally compact space, V a compatible uniformity
and ∆2 = K(Y ). Then the corresponding ∆2-AW filter V∆2 on 2Y is a uni-
formity (see [4] or [5]) and it will be denoted with U(F). Moreover, if 2Y is
equipped with the Fell topology τ2(F), it is known that U(F) is compatible
with τ2(F) (see [4] and [10]).

Observe that in this case (2Y ,τ2(F)) is a compact Hausdorff space and thus
U(F) is the unique uniformity on 2Y corresponding to the Fell topology and
it is generated by all τ2(F) × τ2(F) open neighbourhoods of the diagonal in
2Y × 2Y .

Thus, if Y is a Tychonoff space with a compatible uniformity V, ∆2 ⊂
CL(Y ) and VH and V∆2 the associated HB-uniformity and ∆2-AW filter on
2Y respectively, then on the space F:

(a) A typical basic open set in the HB-uniform convergence topology
τ(UC∆1,VH) on ∆1 is of the form

< f,A,VH >= {g ∈ F : for all x ∈ A, (f(x),g(x)) ∈ VH}, where
f ∈ F, A ∈ ∆1 and VH ∈VH.

If ∆1 = K(X), we get one of the most important topologies, namely
the HB-uniform convergence topology on compacta τ(UCC,VH).

If we replace A by X, we get another important topology: the HB-
uniform convergence topology τ(UC,VH).

If ∆1 is the family of all finite subsets of X, then we have the point-
wise HB-convergence topology τp(VH).

When VH is understood, we may omit it and just write τ(UCC) and τ(UC).

(b) The topology on F generated by



450 G. Di Maio, E. Meccariello and S. Naimpally

{< f,A,M >: f ∈ F,A ∈ ∆1 and M ∈V∆2}
is called the ∆2-AW convergence topology τ(UC∆1,V∆2 ) on

∆1.
If A = X, we have the ∆2-AW convergence topology τ(UC,V∆2 ).

As before, we replace ∆1 by C for ”compacta”.
If ∆1 = K(X), we obtain the ∆2-AW convergence topology on

compacta τ(UCC,V∆2 ).
If ∆1 is the family of all finite subsets of X, then we have the point-

wise ∆2-AW convergence topology τp(V∆2 ).

By Remark 1.2 it follows that whenever Y is a locally compact space and
2Y is equipped with the Fell topology τ2(F), then the corresponding ∆2-AW
filter U(F) is a uniformity which is independent of the uniformity V chosen on
Y .

Note that the topology τ(UCC,U(F)) on F is just what McCoy calls ”Fell
uniform topology (on compact sets)” (see [24]).

(6) Pseudo uniform topologies:

In [22] and [25] function space topologies akin to uniform topologies were
studied. The range space was not necessarily uniformizable. Here we introduce
a similar concept. Let W be a symmetric neighbourhood of the diagonal in
(2Y × 2Y ,τ2 × τ2).

For each f ∈ F and A ∈ ∆1 we set
W?(f,A) = {g ∈ F : for all x ∈ A, (f(x),g(x)) ∈ W}.

The topology on F generated by {W?(f,A) : f ∈ F,A ∈ ∆1 and W a
symmetric τ2×τ2 neighbourhood of the diagonal in 2Y ×2Y} is the τ2-pseudo
uniform topology on ∆1: ps(τ(UC∆1,τ2)).

If A = X, we have the pseudo τ2-uniform topology ps(τ(UC,τ2)).
As before, if ∆1 = K(X), we replace ∆1 by C for ”compacta” and we have

the pseudo τ2-uniform topology on compacta ps(τ(UCC,τ2)).

In case (2Y ,τ2) is uniformizable and we restrict W ’s to symmetric en-
tourages, we do get a uniform topology. This is true as in (5) above or in
(6) when Y is a locally compact space and 2Y is equipped with the Fell topol-
ogy τ2(F) on 2Y and in this case ps(τ(UC∆1,τ2(F)) = τ(UC∆1,U(F)) (cf.
above Remark 1.2).

Although McCoy got his results in uniform setting, we find that some of his
results do not need uniformity at all!

Finally, if τ2 is a given hypertopology on 2Y , then τp(τ2) is the corresponding
τ2-pointwise convergence topology on F which agrees with the pseudo τ2
uniform topology on ∆1 ps(τ(UC∆1,τ2)) when ∆1 is the family of all finite
subsets of X.



Graph topologies on closed multifunctions 451

Those interested in more details are referred to [4] for hypertopologies, [30]
for proximities, [26] and [28] for function space topologies, [9], [10] and [24] for
uniform topologies and convergences on spaces of multifunctions.

2. Basic results.

One of the most valuable result in function space topologies is the embed-
ding of the range space in the function space (cf. Theorem 2.1.1, page 15 in
[26]). In this section we prove similar results for multifunctions which are of
fundamental importance in our work. We need to introduce ”upper” hyper-
topologies that are specially meant for the family C = {X ×E : E ∈ CL(Y )}
of constant multifunctions. These topologies depend on ∆2 alone, unlike
other hypertopologies which depend on either ∆ or ∆1 ×∆2. We use the suffix
r (for range) for such topologies.

On CL(Z), we have the upper r-∆2-topology τ(r∆2)+ which is generated
by the basis {(X ×V )+ : V c ∈ ∆2}∪{CL(Z)}.

Similarly, we have the upper r-∆2U-topology τ(r∆2U)+ which is gene-
rated by the basis {(X ×V )+ : V c ∈ ∆2 or clV ∈ ∆2}∪{CL(Z)}.

If δ2 is a LO-proximity on Y , then we define an associated ”proximity” rδ2
on C by (X ×E) � (X ×V ) w.r.t. rδ2 iff E � V w.r.t. δ2.

If δ2 is an EF-proximity on Y , then it is easy to see that rδ2 on C is also
EF.

Naturally we also have the proximal versions:

the upper proximal r-∆2-topology σ(r∆2,rδ2)+;
the upper proximal r-∆2U-topology σ(r∆2U,rδ2)+.

We have on CL(Z)

the r-∆2-topology τ(r∆2) = τ(r∆2)+ ∨ τ−V .

Similarly the analogues:

the r-∆2U-topology τ(r∆2U) = τ(r∆2U)+ ∨ τ−V ;
the proximal r-∆2-topology σ(r∆2,rδ2) = σ(r∆2,rδ2)+ ∨ τ−V and
the proximal r-∆2U-topology σ(r∆2U,rδ2) = σ(r∆2U,rδ2)+ ∨ τ−V .

Let P(Y ) and P(Z) denote the set of all subsets of Y and Z, respectively.
Consider the map j : P(Y ) ↪→ P(Z) defined by j(E) = (X × E) ∈ P(Z) .
Obviously,

(a) j : CL(Y ) ↪→C is a bijection;



452 G. Di Maio, E. Meccariello and S. Naimpally

(b) j(V +) ⊂ [j(V )]+;
(c) j(V ++δ2 ) ⊂ [j(V )]

++
rδ2

;
(d) j(V −) ⊂ [j(V )]−;
(e) j(E) ∈ W− and W ∈ τ together imply E ∈ [p2(W)]−, where p2 : Z →

Y is the projection.
(f) Let ∆1 ⊂ CL(X), D ∈ ∆1, U a filter on 2Y , M a symmetric member

of U, A ∈ CL(Y ) and f = j(A) = X ×A. Then:
< f,x,M > ∩C =< f,D,M > ∩C =< f,X,M > ∩C = j(M(A))
for each x ∈ X.

So, we have the following results.

Theorem 2.1. Let X and Y be Hausdorff spaces with compatible LO-proximities.
The following are embeddings:

(a) j : (CL(Y ),τ2(∆2)+) ↪→ (CL(Z),τ(r∆2)+);
(b) j : (CL(Y ),τ2(∆2U)+) ↪→ (CL(Z),τ(r∆2U)+);
(c) j : (CL(Y ),σ2(∆2,δ2)+) ↪→ (CL(Z),σ(r∆2,rδ2)+);
(d) j : (CL(Y ),σ2(∆2U,δ2)+) ↪→ (CL(Z),σ(r∆2U,rδ2)+);
(e) j : (CL(Y ),τ2(∆2)) ↪→ (CL(Z),τ(r∆2));
(f) j : (CL(Y ),τ2(∆2U)) ↪→ (CL(Z),τ(r∆2U));
(g) j : (CL(Y ),σ2(∆2,δ2)) ↪→ (CL(Z),σ(r∆2,rδ2));
(h) j : (CL(Y ),σ2(∆2U,δ2)) ↪→ (CL(Z),σ(r∆2U,rδ2)).

Remark 2.2. Let Y be a Tychonoff space, V a compatible uniformity on Y ,
∆1 ⊂ CL(X) and ∆2 ⊂ CL(Y ). If VH and V∆2 are respectively the associated
HB-uniformity and ∆2-AW filter on 2Y , then on C:

(1) τ(UC,VH) = τ(UC∆1,VH) = τp(VH);
(2) τ(UC,V∆2 ) = τ(UC∆1,V∆2 ) = τp(V∆2 ).

Thus the following are embeddings:
(1a) j : (CL(Y ),τ2(VH)) ↪→ (CL(Z),τ(UC,VH));
(1b) j : (CL(Y ),τ2(VH)) ↪→ (CL(Z),τ(UC∆1,VH)).
(2a) j : (CL(Y ),V∆2 ) ↪→ (CL(Z),τ(UC,V∆2 ));
(2b) j : (CL(Y ),V∆2 ) ↪→ (CL(Z),τ(UC∆1,V∆2 )).
Similarly, if Y is a Hausdorff space, τ2 a given hypertopology on 2Y and

∆1 ⊂ CL(X), then on C:
(3) ps(τ(UC,τ2)) = ps(τ(UC∆1,τ2)) = τp(τ2).

Thus the following are embeddings:
(3a) j : (CL(Y ),τ2) ↪→ (CL(Z),ps(τ(UC,τ2)));
(3b) j : (CL(Y ),τ2) ↪→ (CL(Z),ps(τ(UC∆1,τ2))).

Lemma 2.3. Let X and Y be Hausdorff spaces. Then, on the family C of con-
stant multifunctions:

(a) τ(w∆)+ ≤ τ(r∆2)+ ≤ τ(r∆2U)+ ≤ τ(V )+;
and if p2(∆) ⊂ ∆2, then
τ(w∆)+ ≤ τ(∆)+ ≤ τ(r∆2)+ ≤ τ(r∆2U)+ ≤ τ(V )+.



Graph topologies on closed multifunctions 453

(b) τ(w∆)+ ≤ τ(∆)+ ≤ τ(∆U)+ ≤ τ(V )+;
and if p2(∆) ⊂ ∆, then
τ(w∆)+ ≤ τ(∆)+ ≤ τ(∆U)+ ≤ τ(r∆2U)+ ≤ τ(V )+.

(c) τ(w∆) ≤ τ(r∆2) ≤ τ(r∆2U) ≤ τ(V );
and if p2(∆) ⊂ ∆2, then
τ(w∆) ≤ τ(∆) ≤ τ(r∆2) ≤ τ(r∆2U) ≤ τ(V ).

(d) τ(w∆) ≤ τ(∆) ≤ τ(∆U) ≤ τ(V );
and if p2(∆) ⊂ ∆2, then
τ(w∆) ≤ τ(∆) ≤ τ(∆U) ≤ τ(r∆2U) ≤ τ(V ).

Lemma 2.4. Let X and Y be Hausdorff spaces with compatible LO-proximi-
ties. Then, on the family C of constant multifunctions:

(a) σ(w∆)+ ≤ σ(r∆2)+ ≤ σ(r∆2U)+ ≤ σ+;
and if p2(∆) ⊂ ∆2, then
σ(w∆)+ ≤ σ(∆)+ ≤ σ(r∆2)+ ≤ σ(r∆2U)+ ≤ σ+.

(b) σ(w∆)+ ≤ σ(∆)+ ≤ σ(∆U)+ ≤ σ+;
and if p2(∆) ⊂ ∆2, then
σ(w∆)+ ≤ σ(∆)+ ≤ σ(∆U)+ ≤ σ(r∆2U)+ ≤ σ+.

(c) σ(w∆) ≤ σ(r∆2) ≤ σ(r∆2U) ≤ σ;
and if p2(∆) ⊂ ∆2, then
σ(w∆) ≤ σ(∆) ≤ σ(r∆2) ≤ σ(r∆2U) ≤ σ.

(d) σ(w∆) ≤ σ(∆) ≤ σ(∆U) ≤ σ;
and if p2(∆) ⊂ ∆2, then
σ(w∆) ≤ σ(∆) ≤ σ(∆U) ≤ σ(r∆2U) ≤ σ.

We say that Z is locally ∆ iff for each z ∈ Z with z ∈ V ∈ τ, there is
D ∈ ∆ with z ∈ intD ⊂ D ⊂ V . (Note that this is a generalization of local
compactness in which case ∆ = K(Z)).

Lemma 2.5. Let X and Y be Hausdorff spaces with compatible LO-proximi-
ties, Z = X ×Y and ∆ ⊂ CL(Z) a cover. If Z is locally ∆, then:

(a) τ(∆)+ = τ(∆U)+ if and only if Z ∈ ∆ i.e. ∆ = CL(Z).
(b) σ(∆)+ = σ(∆U)+ if and only if Z ∈ ∆ i.e. ∆ = CL(Z).

Proof. We prove only (a). It suffices to show that τ(∆U)+ ≤ τ(∆)+ implies
Z ∈ ∆. Suppose U is a nonempty subset of Z with A ⊂ U and clU ∈ ∆ (note
that such U exists since Z is locally ∆). Then there is an open subset V in Z
with V c ∈ ∆ such that A ⊂ V ⊂ U. Clearly Z = clU ∪V c ∈ ∆. �

Corollary 2.6. Let X and Y be Hausdorff spaces with compatible LO-proxi-
mities, Z = X ×Y and ∆ ⊂ CL(Z) a cover. If Z is locally ∆, then:

(a) τ(∆) = τ(∆U) if and only if Z ∈ ∆ i.e. ∆ = CL(Z) (cf. [14], Theorem
3.2).

(b) σ(∆) = σ(∆U) if and only if Z ∈ ∆ i.e. ∆ = CL(Z).
(c) When ∆ = K(Z), we have τ(F) = τ(U) if and only if X is compact.

Remark 2.7. In the following relations, vertical lines show embeddings:



454 G. Di Maio, E. Meccariello and S. Naimpally

(a)
CL(Y ): τ2(∆2) ≤ τ2(∆2U) ≤ τ2(V )
↓ j ↓ ↓ ↓
C ⊂ F: τ(w∆) ≤ τ(r∆2) ≤ τ(r∆2U) ≤ τ(rCL(Y )) ≤ τ(V ).

(b)
CL(Y ): τ2(∆2) ≤ τ2(∆2U) ≤ τ2(V )
↓ j ↓ ↓
C ⊂ F: τ(w∆) ≤ τ(∆) ≤ τ(∆U) ≤ τ(r∆2U) ≤ τ(rCL(Y )) ≤ τ(V ).

(c)
CL(Y ): σ2(∆2) ≤ σ2(∆2U) ≤ σ2 ≤ τ2(VH)
↓ j ↓ ↓ ↓ ↓
C ⊂ F: σ(w∆) ≤ σ(∆) ≤ σ(r∆2) ≤ σ(r∆2U) ≤ σ(rCL(Y )) ≤ τ(VH).

(d)
CL(Y ): σ2(∆2) ≤ σ2(∆2U) ≤ σ2 ≤ τ2(VH)
↓ j ↓ ↓ ↓
C ⊂ F: σ(w∆) ≤ σ(∆) ≤ σ(∆U) ≤ σ(r∆2U) ≤ σ(rCL(Y )) ≤ τ(VH).

(e) If the proximities involved are EF, then

CL(Y ): σ2(∆2) ≤ σ2(∆2U) ≤ σ2 ≤ τ2(V )
↓ j ↓ ↓ ↓ ↓
C ⊂ F : σ(w∆) ≤ σ(∆) ≤ σ(r∆2) ≤ σ(r∆2U) ≤ σ(rCL(Y )) ≤ τ(rCL(Y )) ≤

τ(V ).

(f)
CL(Y ): σ2(∆2) ≤ σ2(∆2U) ≤ σ2 ≤ τ2(V )
↓ j ↓ ↓ ↓
C ⊂ F : σ(w∆) ≤ σ(∆) ≤ σ(r∆2) ≤ σ(r∆2U) ≤ σ(rCL(Y )) ≤ τ(rCL(Y )) ≤

τ(V ).
(g)
CL(Y ): τ2(∆2) ≤ τ2(V )
↓ j ↓ ↓
C ⊂ F: ps(τ(UC∆1,τ2(∆2))) ≤ τ(rCL(Y )) ≤ τ(V ).

3. Generalization of McCoy’s Results.

In this section we begin comparing some of the topologies defined in the
previous section and find conditions for their pairwise equivalence. We study
some simple ones which are analogues of those in McCoy’s paper and state
McCoy’s results just below the analogues. Again, we recall that (X,τ1) and
(Y,τ2) are Hausdorff spaces. We set Z = X×Y and assign the product topology
τ = τ1 × τ2. If δ1 and δ2 are compatible proximities on X and Y respectively,
then on Z is assigned the product proximity δ = δ1 × δ2. The family 2Z of



Graph topologies on closed multifunctions 455

closed subsets of Z can be identified with the space F of all set valued maps
on X to 2Y taking points of X to closed (possibly empty) subsets of Y . Other
assumptions will be stated at the places where they are needed.

McCoy assumed that both X and Y are locally compact spaces and Y is
a non-trivial complete metric space. He then studied four topologies: τ(F),
τ(V ), τ(UCC,U(F)) and τ(VH). In this section we pursue τ(∆), τ(w∆), σ(∆),
σ(w∆), τ(V ), τ(VH), τ(UC∆1,VH) and τ(UC,VH) as well as τ(UC∆1,V∆2 )
and τ(UC,V∆2 ). Moreover, we also consider ps(τ(UC,τ2)) and ps(τ(UC∆1,τ2))
for an arbitrary topology τ2 on 2Y .

Theorem 3.1. Let X and Y be Hausdorff spaces with compatible LO-proximities
δ1 and δ2, respectively. Then on F:

(a) τ(w∆)+ = τ(∆1 × ∆2)+ ≤ τ(∆)+ ≤ τ(∆U)+ ≤ τ(V )+.
(b) τ(w∆) ≤ τ(∆) ≤ τ(∆U) ≤ τ(V ).

Moreover, if δ = δ1 × δ2 is EF , then:
(c) σ(w∆,δ)+ ≤ σ(∆,δ)+ ≤ σ(∆U,δ)+ ≤ σ(δ)+ ≤ τ(V )+.
(d) σ(w∆,δ) ≤ σ(∆,δ) ≤ σ(∆U,δ) ≤ σ(δ) ≤ τ(V ).

(Cf. [24] Prop. 4.1) If X and Y are Hausdorff spaces, then
τ(F) ≤ τ(U) ≤ τ(V ).

The following results and Lemmas play a key role.

Lemma 3.2. Let X be a Hausdorff space, Y a Tychonoff space, U and V
compatible uniformities on Y with U ⊂ V and UH and VH the corresponding
HB-uniformities on 2Y associated to U and V, respectively. Then on F:

(a) τ(UC∆1,UH) ≤ τ(UC∆1,VH);
(b) τ(UC,UH) ≤ τ(UC,VH).

Furthermore, if ∆2 ⊂ CL(Y ) and U∆2 and V∆2 are the corresponding ∆2-
AW filters on 2Y associated to U and V, respectively, then:

(c) τ(UC∆1,U∆2 ) ≤ τ(UC∆1,V∆2 );
(d) τ(UC,U∆2 ) ≤ τ(UC,V∆2 ).

Proof. (a) and (b) (resp. (c) and (d)) follow from the fact that if U ⊂V on Y ,
then UH ⊂VH (resp. U∆2 ⊂V∆2 ) on 2Y . �

Lemma 3.3. Let Y be a Tychonoff space, V a compatible uniformity on Y ,
∆2 ⊂ CL(Y ), VH and V∆2 the corresponding HB-uniformity and ∆2-AW filter
on 2Y , respectively. Then, on 2Y , V∆2 ⊂VH.

Proof. Let [D,V ] be basic element of V∆2 where D ∈ ∆ and V ∈ V. We
claim that the corresponding element VH ∈ VH is such that VH ⊂ [D,V ].
Assume not. Then there exists (A,B) ∈ 2Y × 2Y such that (A,B) ∈ VH but
(A,B) 6∈ [D,V ]. Thus either (i) A ∩ D 6⊂ V (B) or (ii) B ∩ D 6⊂ V (A). If (i)
occurs, then there exists y ∈ (A∩D)\V (B); a contradiction because A ⊂ V (B).
Similarly, if (ii) occurs. �



456 G. Di Maio, E. Meccariello and S. Naimpally

Corollary 3.4. Let X be a Hausdorff space, Y a Tychonoff space, V a com-
patible uniformity on Y , ∆1 ⊂ CL(X), ∆2 ⊂ CL(Y ), VH and V∆2 the corre-
sponding HB-uniformity and ∆2-AW filter on 2Y . Then on F:

(a) τ(UC∆1,V∆2 ) ≤ τ(UC∆1,VH);
(b) τ(UC,V∆2 ) ≤ τ(UC,VH).

Proof. (a) and (b) follow from above Lemma 3.3. �

We recall that if (Z,τ) is a Tychonoff space with a compatible EF-proximity
δ, then a uniformity U on Z is called compatible w.r.t. δ iff the uniform pro-
ximity δ(U) induced by U equals δ (see Section 1 and [30]). δ admits a unique
compatible totally bounded uniformity Uw ([30]).

Theorem 3.5. (Cf. [18]) Let Y be a Tychonoff space and δ2 a compatible
EF -proximity on Y . The corresponding proximal topology σ2(δ2) on 2Y is
always uniformizable. In fact, it is the topology induced on 2Y by the HB-
uniformity UHw which is derived from the unique totally bounded uniformity Uw
on Y compatible w.r.t. δ2.

Lemma 3.6. (Cf. Theorem 2.1 in [16]) Let Y be a Tychonoff space, δ2 a
compatible EF -proximity on Y and ∆2 ⊂ CL(Y ) a cover. If the proximal ∆2
topology σ2(∆2,δ2) is uniformizable, then the proximal ∆2-topology σ2(∆2,δ2)
equals the topology τ(U∆w ) induced by the ∆-AW filter U∆2w , where Uw is the
unique totally bounded uniformity on Y compatible w.r.t. δ2.

Proof. Let Uw be the unique totally bounded uniformity on Y compatible with
δ2. Without loss of generality we assume that all entourages W ∈ Uw are
open and symmetric.

First, let {Aλ : λ ∈ Λ} ⊂ 2Y be a net τ(U∆2w )-converging to A ∈ 2Y . We
claim that the net {Aλ : λ ∈ Λ} σ2(∆2,δ2)-converges to A.

(i) If A ∈ V − where V ∈ τ, then there exist a ∈ A∩V and a W ∈Uw such
that W(a) ⊂ V . Since A ∈ [{a},W ](A) ⊂ V −, eventually
Aλ ∈ [{a},W ](A) ⊂ V −.
(ii) If A ∈ (Dc)++δ2 where D ∈ ∆2, then D �δ2 A

c. Since Uw is compatible
w.r.t. δ2 and by assumption σ2(∆2,δ2) is uniformizable there are S ∈ ∆2 and
W ∈Uw such that D ⊂ W(D) ⊂ S ⊂ W(S) ⊂ Ac (see Theorem 4.4.5, Lemma
4.4.3 and Definition 4.4.2 in [4]). Since A ∈ [S,W](A), W(A) ∩ S = ∅ and
eventually Aλ ∈ [S,W ](A), then eventually Aλ ∈ (Dc)++δ2 .

Thus σ2(∆2,δ2) ≤ τ(U∆2w ).

On the other hand, let {Aλ : λ ∈ Λ} ⊂ 2Y be a net σ2(∆2,δ2)-converging
to A ∈ 2Y . We claim that the net {Aλ : λ ∈ Λ} τ(U∆2w )-converges to A. So,
let [W,D](A) a τ(U∆w )-neigbourhood at A where D ∈ ∆2 and W ∈ Uw. Let
V ∈Uw be such that V 2 ⊂ W . We have two cases:

(i) A ∈ (Dc)++δ2 . Then eventually Aλ ∈ (D
c)++δ2 and obviously, ∅ = Aλ∩D ⊂

W(A) and ∅ = A∩D ⊂ W(Aλ), i.e eventually Aλ ∈ [W,D][A].



Graph topologies on closed multifunctions 457

(ii) A 6∈ (Dc)++δ2 . Then V (A) ∩ D 6= ∅. Since V is totally bounded, there

are xj ∈ A, 1 ≤ j ≤ n such that A ⊂
n⋃
j=1

V (xj) ⊂ V 2(A). Since A∩V (xj) 6= ∅

for each j, eventually Aλ ∩ V (xj) 6= ∅ and so xj ∈ V (Aλ). Hence, eventu-

ally A ∩ D ⊂
n⋃
j=1

V (xj) ⊂ V 2(Aλ) ⊂ W(Aλ). Note that (D ∩ V (A)c) ∈ ∆2

and A ∈ (Dc ∪ V (A))++δ2 ∈ σ2(∆,δ2). So eventually, Aλ ∈ (D
c ∪ V (A))++δ2 .

Thus eventually Aλ ∩ D = [Aλ ∩ (D ∩ V (A))] ⊂ W(A), i.e. eventually
Aλ ∈ [W,D](A). Thus τ(U∆w ) ≤ σ2(∆2,δ2). Combining the earlier part we get
τ(U∆2w ) = σ2(∆2,δ2). �

Remark 3.7.
(a) In [15] it is shown that if τ2(∆2) is uniformizable, then there is a

compatible EF-proximity δ2 on Y such that τ2(∆2) = σ2(∆2,δ2) (see
Lemma 2.2 in [15]).

(b) Above Lemma and Remark 3.7 (a) show that the appropriate exten-
sion of U(F) (see Remark 1.2) for uniformizable ∆2- and proximal ∆2-
topologies are the ∆2-AW filters induced by compatible totally bounded
uniformities Uw on Y . Thus, as in the definition of U(F) (see Remark
1.2), we reserve the symbol U(τ2) to denote the corresponding compat-
ible AW filter associated with the totally bounded uniformity Uw on Y
whenever τ2 is a uniformizable (proximal) ∆2-topology on 2Y .

Theorem 3.8. Let X be a Hausdorff space, Y a Tychonoff space with a com-
patible EF -proximity δ2 and Uw the unique totally bounded uniformity asso-
ciated to δ2, ∆1 ⊂ CL(X), ∆2 ⊂ CL(Y ) a cover and 2Y equipped with the
proximal ∆2-topology σ2(∆2) induced by δ2. If σ2(∆2) is uniformizable and
U(σ2(∆2)) and UH are respectively the corresponding ∆2-AW filter compatible
w.r.t. σ(∆2) and HB-uniformity on 2Y associated to Uw, then on F:

(a) τ(UC∆1,U(σ2(∆2))) ≤ τ(UC∆1,UH);
(b) τ(UC,U(σ2(∆2))) ≤ τ(UC,UH).

Proof. (a) and (b) follow by Corollary 3.4 and Lemma 3.6. �

Next Theorem generalizes Propositions 4.2 and 4.5 and shows that the as-
sumption of local compactness on the base space X it is not nedded (see also
Proposition 4.1 in [10]).

Theorem 3.9. Let X be a Hausdorff space, Y a Tychonoff space, V a compati-
ble uniformity on Y and VH the corresponding HB-uniformity on 2Y . Suppose
δ2 is an EF -proximity on Y with δ2 ≤ δ2(V), 2Y equipped with the proxi-
mal topology σ2 induced by δ2. If U(σ2) is the corresponding compatible HB-
uniformity associated with Uw, the unique totally bounded uniformity compatible
w.r.t. δ2, then on F:

(a) τ(UC∆1,U(σ2)) ≤ τ(UC∆1,VH).



458 G. Di Maio, E. Meccariello and S. Naimpally

(b) τ(UC,U(σ2)) ≤ τ(UC,VH).
(c) (Cf. Prop. 4.2 in [24] and Prop. 4.1 in [10]) If X is a Hausdorff

space, Y a locally compact space and τ2(F) the Fell topology on 2Y ,
then τ(UCC,U(F)) ≤ τ(UCC,VH).

(d) Furthermore, there is equality either in (a) or in (b) if and only if V is
totally bounded.

(e) (Cf. Prop. 4.5 in [24]) If X is a Hausdorff space, Y a locally com-
pact and completely metrizable space with metric d, V the metric uni-
formity associated with d and τ2(F) the Fell topology on 2Y , then
τ(UCC,U(F)) = τ(UCC,VH) if and only if Y is compact.

Proof. Let Uw and U′w be the totally bounded uniformities on Y compatible
with δ2 and δ2(V), respectively. From Theorems 12.7 and 12.14 in [30] Uw ⊂U′w
and U′w ⊂V and hence Uw ⊂V.

Thus (a) and (b) follow from Theorem 3.5 and (a) and (b) in Lemma 3.2.
(d) It follows from the fact that equality either in (a) or in (b) is equivalent

to σ2(δ2) = τ2(VH), which in turn, is equivalent to the total boundedness of
V. Whereas (c) and (e) follow from above (a) and (d) respectively when ∆1 =
K(X), Remark 1.2 and the well-known relation τ2(F) = σ2(F,δ2) ≤ σ2(δ2). �

Next Theorem shows that in Propositions 4.6 and 4.8 in [24] the assumption
of local compactness on the base space X can be dropped. First we give the
following Remark.

Remark 3.10. It is well known (see [32]) that on fuction spaces the lower
Vietoris topology is coarser than the topology of pointwise convergence. Thus,
whenever τ2 is a given topology on 2Y , we have:

(?) τ(V −) ≤ τp(τ2).

Theorem 3.11. Let X be a Hausdorff space with a compatible LO- proximity
δ1, Y a Tychonoff space with a compatible EF -proximity δ2, ∆1 ⊂ CL(X),
∆2 ⊂ CL(Y ) a cover, Z = X×Y equipped with the product proximity δ = δ1×δ2
and 2Y equipped with the proximal ∆2-topology σ2(∆2) induced by δ2. If σ2(∆2)
is uniformizable and U(σ2(∆2)) is the corresponding compatible ∆2-AW filter
associated to Uw, the unique totally bounded uniformity compatible w.r.t. δ2,
then on F:

(a) σ(w∆,δ) ≤ τ(UC∆1,U(σ2(∆2))) ≤ τ(UC,U(σ2(∆2))).
(b) Furthermore, under the conditions of Theorem (3.9) we have

σ(w∆,δ) ≤ τ(UC∆1,U(σ2(∆2)) ≤ τ(UC∆1,VH).
(Cf. [24] Prop. 4.3 and Prop. 4.4) If X is a Hausdorff space, Y a
locally compact space and V a compatible uniformity on Y , then
τ(F) ≤ τ(UCC,U(F)) ≤ τ(UCC,VH).

Proof. First we show (a). So, let M = U × V , U ∈ τ1, V ∈ τ2 and f ∈ M−.
Hence, there is a point (x,y) ∈ f ∩M. Then by (?) in the above Remark there
exists a τp(σ2(∆2))-neighbourhood H of f such that H⊂ M− and clearly H is
also a τ(UC∆1,U(σ2(∆2))-neighbourhood of f.



Graph topologies on closed multifunctions 459

Next, suppose D = A × B where A ∈ ∆1, B ∈ ∆2 and f �δ Dc. Since
δ = δ1 × δ2 and δ2 is EF, there is an open set V in Y with B �δ2 V and
f �δ (A×V c).

Let Uw be the unique totally bounded uniformity on Y compatible with δ2.
Thus, there is a W ∈Uw such that B ⊂ W(B) ⊂ W 2(B) ⊂ V (see [30]).

Clearly, < f,A; [B,W ] > is τ(UC∆1,U(σ2(∆2))-neighbourhood of f. We
claim < f,A; [B,W ] >⊂ (Dc)++δ . In fact, if g ∈< f,A; [B,W ] >, then g(x) ∈
[B,W ] for each x ∈ A. Now, W 2(B) ∩ V c = ∅ together with g(x) ∈ [B,W ]
for each x ∈ A imply g �δ A×V c. Hence g ∈ (Dc)++δ . So the first inclusion
follows. The second one is trivial.

(b) It follows from (a) above and Corollary 3.4. �

Remark 3.12. By (a) in Remark 3.7 we give statements and proofs only for
the τ(UC∆1,U(σ2(∆2))) topology. Similar ones for the τ(UC∆1,U(τ2(∆2)))
topology, when τ2(∆2) is uniformizable and ∆2 is a cover, are left to the reader.

Theorem 3.13. Let X be a Hausdorff space with a compatible LO-proximity
δ1, Y a Tychonoff space with a compatible EF -proximity δ2, Z = X × Y
equipped with the product proximity δ = δ1 × δ2, ∆2 ⊂ CL(Y ) a cover and
2Y equipped with the proximal ∆2-topology σ2(∆2) induced by δ2. Let σ2(∆2)
be uniformizable and U(σ2(∆2)) the corresponding compatible ∆2-AW filter.
Then on F:

(a) If ∆1 is the family of all finite subsets of X, then
τ(UC∆1,U(σ2(∆2))) ≤ σ(w∆).

(b) If τ(UC∆1,U(σ2(∆2))) ≤ σ(w∆), then X is discrete.
(Cf. [24] Prop. 4.6 and Prop. 4.8) If X is a Hausdorff space and Y a
locally compact space, then the following are equivalent:

(α) X is discrete;
(β) τ(F) = τ(UCC,U(F));
(γ) τ(UCC,U(F)) ≤ τ(F) ≤ τ(UCC,VH).

(c) Let Y be a Tychonoff space, V a compatible uniformity on Y and VH the
corresponding HB-uniformity on 2Y . Then the HB- uniform topology
on ∆1 τ(UC∆1,VH) equals the weak proximal ∆ topology σ(w∆) if
and only if each member of ∆1 is finite and V is totally bounded.
([24] Prop. 4.7) If Y is a completely metrizable space with metric d and
V is the d-metric uniformity, then τ(UCC,VH) = τ(F) if and only if
X is discrete and Y is compact.

Proof. (a) It suffices to observe that if ∆1 is the family of all finite subsets of
X, then τ(UC∆1,U(σ2(∆2))) = τp(σ2(∆2)) and clearly τp(σ2(∆2)) ≤ σ(w∆).

(b) Suppose X is not discrete. Then there exists a point x0 in X which is not
isolated. Denote by N(x0) the family of all open neighbourhoods of x0 and let
y0, y1 be two distinct points in Y . For U ∈N(x0) define fU (x) = {y0,y1} for
x 6∈ U and fU (x) = {y0} for x ∈ U. It is easy to verify that fU ∈ F and that the
net {fU : U ∈N(x0)} σ(w∆)-converges to a multifunction f defined by f(x) =
{y0,y1} for x ∈ X. Since {fU (x0) : U ∈ N(x0)} does not τ2(V −) converge to



460 G. Di Maio, E. Meccariello and S. Naimpally

f(x0), it follows that {fU : U ∈N(x0)} cannot τ(UC∆1,U(σ2(∆2))) converge
to f.

(c) It follows from the above and the fact that the equality is equivalent
to σ2(∆2) = τ2(VH), which in turn, is equivalent to the total boundedness of
V. �

Theorem 3.14. Let X be a Hausdorff space, Y a Tychonoff space, V a com-
patible uniformity on Y and VH the corresponding HB-uniformity on 2Y . Then
on C:

(a) If V is totally bounded, then τ(UC,VH) ≤ τ(rCL(Y )) ≤ τ(V ).
(b) If V is not totally bounded, then on C, τ(UC,VH)6≤τ(V ).

Thus, if on F τ(UC,VH) ≤ τ(V ), then V is totally bounded.
([24] Prop. 4.9) If X is a locally compact space, Y a locally compact
completely metrizable space with metric d and V the d-metric unifor-
mity, then on F τ(UCC,VH) ≤ τ(V ) if and only if X is discrete and
Y is compact.

Proof. (a) Let U ∈ V be open and symmetric and f = (X × E) ∈ C. Total
boundedness of V implies there is a finite set {yk : 1 ≤ k ≤ n}) ⊂ Y such

that E ⊂
n⋃
k=1

U(yk). Consider a typical τ(UC,VH)- neighbourhood of f, i.e.

< f,X,U2 >. It is easy to see that [X × U(E)]+ ∩
n⋂
k=1

U(yk)
− is a τ(wV )-

neighbourhood of f which is contained in < f,X,U2 > .

(b) If V is not totally bounded there is an open U ∈ V and a sequence

{yk : k ∈ IN} ⊂ Y such that Y 6⊂
n⋃
k=1

U(yk) for each n ∈ IN. Then it is clear

that FC = {f = X ×E : E ⊂ Y is finite} is a subset of C which is not dense in
(C,τ(UC,VH)) but which is dense in (C,τ(V )). �

Proposition 3.15. Let X be a Hausdorff space, Y a Tychonoff space, V a
compatible uniformity on Y and VH the corresponding HB- uniformity on 2Y .
If on F τ(V ) ≤ τ(UC∆1,VH), then X ∈ ∆1 and Y is Atsuji (i.e. ∆1 = CL(X)
and every real-valued continuous function on Y is uniformly continuous).

Proof. First we show X ∈ ∆1. Assume not and let y1, y2 be distinct points
of Y . Define f = X ×{y1} and D = X ×{y2}. Then f ∈ (Dc)+ but for
any A ∈ ∆1 and any V ∈ V, < f,A; VH > is not contained in (Dc)+. In
fact, choose x′ ∈ (X \ A) (which exists since we are assumming X 6∈ ∆1)
and set g = f ∪ {(x′,y2)} then g ∈< f,A; VH >, but g 6∈ (Dc)+ because
(x′,y2) ∈ g(x′) ∩ D; a contradiction. Then, the result follows from the fact
that on C τ(V ) ≤ τ(UC,VH) if and only if τ2(V ) ≤ τ2(VH) on CL(Y ) which
in turn is equivalent to Y being Atsuji. �



Graph topologies on closed multifunctions 461

Next Example suggested by Ľubica Holá shows that the converse is not in
general true.

Example 3.16. Let X = [1, +∞) and Y = [0, 1] subspaces of the real line
and VH the Hausdorff metric uniformity on 2Y . Set f = X ×{0} and for each
natural number n define fn = X ×{

1
n
}. Then the sequence {fn : n ∈ IN}

τ(UC,VH)-converges to f but it fails to τ(V )-converge to f. In fact, take
G = {(x,y) ∈ X ×Y : y <

1
x
}. Then f ∈ G but fn 6∈ G, for each n ∈ IN.

However, if ∆1 = K(X) we have the next result.

Proposition 3.17. Let X be a Hausdorff space, Y a Tychonoff and locally ∆2
space, V a compatible uniformity on Y , then on F τ(V ) ≤ τ(UCC,VH) if and
only if X is compact and Y is Atsuji.

(Cf. [24] Prop. 4.10) If X is a Hausdorff space, Y a locally compact
and completely metrizable space with metric d and V the d-metric uni-
formity, then on F τ(V ) ≤ τ(UCC,VH) if and only if X is compact
and Y is a topological sum of a compact space and a discrete space.

Proof. It is known from [29] that on C(X,Y ), the family of all continuous
functions on X to Y , the graph topology equals the Vietoris topology. More-
over, observe that τ(UCC,VH) on C(X,Y ) equals the compact open topology
τk. Thus, from a result analogous to 2(d) page 14 of [26] it follows that on
C(X,Y ), τ(V ) ≤ τk if and only if X ∈ K(X). The statement then follows
from the known result that Y is Atsuji if and only if τ2(V ) ≤ τ2(VH). �

The proof of the next Proposition is left to the reader.

Proposition 3.18. Let X and Y be Hausdorff spaces. Then on F
τ(∆) = τ(V ) if and only if Z ∈ ∆ i.e. ∆ = CL(Z).

([24] Prop. 4.11) τ(F) = τ(V ) if and only if Z is compact i.e. X and
Y are both compact.

To study comparisons between the pseudo uniform topologies with some
other topologies we give the following Lemma.

Lemma 3.19. Let X be a Hausdorff space, Y a Tychonoff and locally ∆2
space, δ1 a compatible LO-proximity on X, δ2 a compatible EF -proximity on
Y , Z = X × Y equipped with the product proximity δ = δ1 × δ2, τ2(V −) and
σ2(∆2) respectively the lower Vietoris topology and the proximal ∆2-topology
on 2Y . Then on F:

(a) τp(τ2(V −)) ≤ ps(τ(UC∆1,σ2(∆2))) ≤ ps(τ(UC,σ2(∆2))).
If Y is a Hausdorff and locally ∆2 space and 2Y is equipped with the ∆2-

topology τ2(∆2), then:
(b) τp(τ2(V −)) ≤ ps(τ(UC∆1,τ2(∆2))) ≤ ps(τ(UC,τ2(∆2))).



462 G. Di Maio, E. Meccariello and S. Naimpally

Proof. We prove only (a). To show (b) few changes are needed.
Suppose < f,{x},V − > is a τp(τ2(V −)) neighbourhood of f, where V ∈ τ2

and f ∈ F. So there is a point y ∈ f(x) ∩ V . Since Y is locally ∆2, there is
a D ∈ ∆2 such that y ∈ intD ⊂ D ⊂ V . Since the proximity δ2 is EF we
also have y ∈ intD ⊂ D � V . Then W = (V − × V −) ∪ [(Dc)++ × (Dc)++]
is a symmetric neighbourhood of the diagonal in (2Y × 2Y ,σ2 ×σ2). Clearly,
f ∈ W?(f,{x}) ⊂ V −. �

Theorem 3.20. Let X be a Hausdorff space with a compatible LO-proximity
δ1, Y a Tychonoff space with a compatible EF -proximity δ2, Z = X × Y
equipped with the product proximity δ = δ1 × δ2 and 2Y equipped with the
proximal ∆2-topology σ2(∆2) induced by δ2. If Y is locally ∆2, then on F:

(a) σ(w∆,δ) ≤ ps(τ(UC∆1,σ2(∆2))) ≤ ps(τ(UC,σ2(∆2))).
If Y is a Hausdorff and locally ∆2 space and 2Y is equipped with the ∆2

topology τ2(∆2), then:
(b) τ(w∆) ≤ ps(τ(UC∆1,τ2(∆2))) ≤ ps(τ(UC,τ2(∆2))).

Proof. Again it suffices to show (a). By above Lemma and Remark 3.10 it
suffices to show that on F

σ(w∆,δ)+ ≤ ps(τ(UC∆1,σ2(∆2))) ≤ ps(τ(UC,σ2(∆2))).
Thus, suppose D = A × B where A ∈ ∆1, B ∈ ∆2 and f � Dc. There is
an open set V in Y with B �δ2 V and such that f � A × V (because δ2 is
EF). Set S = (V −×V −)∪ [(Bc)++ ×(Bc)++] a symmetric neighbourhood of
the diagonal in (2Y × 2Y ,σ2 ×σ2). Then f ∈ S?(f,A) ⊂ (Dc)++. So the first
inclusion follows. The second one is trivial.

Clearly (b) follows from above with obvious changes. �

Theorem 3.21. Let X and Y be Hausdorff spaces with Y locally ∆2, δ1 a
compatible LO-proximity on X, δ2 a compatible LO-proximity on Y and δ =
δ1×δ2 the product proximity on Z = X×Y . Let σ2 and τ2 denote the proximal
∆2-topology and the ∆2-topology on 2Y , respectively. Then on F:

(a) If ∆1 is the family of all finite subsets of X, then
ps(τ(UC∆1,σ2)) ≤ σ(w∆) and ps(τ(UC∆1,τ2)) ≤ τ(w∆).

(b) If ps(τ(UC∆1,σ2)) ≤ σ(w∆) or ps(τ(UC∆1,σ2)) ≤ τ(w∆), then X is
discrete.

Proof. To check (a) observe that if ∆1 is the family of all finite subsets of X,
then ps(τ(UC∆1,σ2(∆2))) = τp(σ2(∆2)) as well as ps(τ(UC∆1,τ2(∆2))) =
τp(τ2(∆2)) and clearly τp(σ2(∆2)) ≤ σ(w∆) as well as τp(τ2(∆2)) ≤ τ(w∆).

(b) We prove only the second part, i.e if ps(UC∆1,τ2(∆2)) ≤ τ(w∆), then
X is discrete. Assume not. Then there exists a point x0 in X wich is not
isolated. Denote by N(x0) the family of all open neighbourhoods of x0 and let
y0, y1 be two different points in Y . For U ∈N(x0) define fU (x) = {y0,y1} for
x 6∈ U and fU (x) = {y0} for x ∈ U. It is easy to verify that fU ∈ F and that
the net {fU : U ∈ N(x0)} τ(w∆)-converges to f defined by f(x) = {y0,y1}



Graph topologies on closed multifunctions 463

for x ∈ X. Since {fU (x0) : U ∈N(x0)} does not τ2(V −) converge to f(x0), it
follows that {fU : U ∈N(x0)} cannot ps(τ(UC∆1,τ2(∆2))) converge to f. �

Theorem 3.22. Let X and Y be Hausdorff spaces. If Y is locally ∆2, then
on F τ(wV ) ≤ ps(τ(UC∆1,τ2(∆2))) if and only if X ∈ ∆1 and Y ∈ ∆2 (i.e.
∆1 = CL(X) and ∆2 = CL(Y )).

(Cf. [24] Prop. 4.12) If X is a Hausdorff space and Y is a locally
compact space, then τ(V ) ≤ τ(UCC,U(F)) if and only if Z is compact
i.e. X and Y are both compact.

Proof. First observe that from Remark 2.8 (g) it follows that on C τ(wV ) ≤
ps(τ(UC∆1,τ2(∆2))) ⇔ τ2(V ) ≤ τ2(∆2) ⇔ Y ∈ ∆2 i.e. ∆2 = CL(Y ). Next,
by (b) in Theorem 3.20 to show that τ(wV ) ≤ ps(τ(UC∆1,τ(V ))) is equivalent
to X ∈ ∆1 it suffices to prove that the inequality τ(wV ) ≤ ps(τ(UC∆1,τ2(V ))
implies X ∈ ∆1. Assume not. Let y1, y2 be distinct points of Y . Set f =
X ×{y1} and D = X ×{y2}. Clearly, D ∈ CL(X) ×CL(Y ) and f ∈ (Dc)+.
We claim that for each ps(τ(UC∆1,τ2(V ))) neighbourhood W?(f,A) of f there
exists g ∈ W?(f,A) such that g 6∈ (Dc)+. In fact, since A 6= X there exists
some x′ ∈ (X \A). Set g = f ∪{(x′,y2)}. Then g ∈ W?(f,A) but g 6∈ (Dc)+
showing thereby that X must be in ∆1. �

The following Example, due to Ľubica Holá, shows that the uniform Haus-
dorff convergence topology τ(UC,VH) is in general not finer than the pseudo
proximal uniform topology ps(τ(UC,σ2)).

Example 3.23. In the real line with the usual metric d, set X =
⋃

n∈ IN
(

1
n + 1

,
1
n

).

Let Y = X, V the metric uniformity on Y associated to d, Vn = (
1

n + 1
,

1
n

),

ηn =
1
n
−

1
n + 1

and yn ∈ Vn be fixed for each n ∈ IN. Let f : X → Y

defined by f(Vn) = yn. Let W =
⋃
n∈IN

(V −n ×V
−
n ) ∪{∅,∅}. Then W is a

symmetric σ2 ×σ2 open neigbourhood of the diagonal of 2Y ×2Y such that the
corresponding W?(f,X) = {g ∈ F : ∀x ∈ X(f(x),g(x)) ∈ W} 6∈ τ(UC,VH).

Assume not, i.e. W?(f,X) ∈ τ(UC,VH). Then there exists a positive
real ε such that < f,X; ε >⊂ W?(f,X), where < f,X; ε >= {g ∈ F :
Hd(f(x),g(x)) < ε ∀x ∈ X} (here Hd denotes the Hausdorff distance as-
sociated to d). Let η < ε. For each x ∈ X, set g(x) = f(x) + η and let n ∈ IN
be such that ηn < η. Of course g ∈< f,X; ε >, but g 6∈ W?(f,X). In fact
choose xn = yn ∈ Vn. Then f(xn) ∈ Vn, but g(xn) 6∈ Vn.

Acknowledgements. The authors are grateful to Ľ. Holá for her detailed
criticism, advice and friendship.



464 G. Di Maio, E. Meccariello and S. Naimpally

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Received February 2002

Revised August 2002

Giuseppe Di Maio

Seconda Università degli Studi di Napoli, Facoltà di Scienze, Dipartimento di
Matematica, Via Vivaldi 43, 81100 Caserta, Italia

E-mail address : giuseppe.dimaio@unina2.it

Enrico Meccariello

Università del Sannio, Facoltà di Ingegneria, Piazza Roma, Palazzo B. Lu-
carelli, 82100 Benevento, Italia

E-mail address : meccariello@unisannio.it

Somashekhar Naimpally

96 Dewson Street, Toronto, Ontario, M6H 1H3, Canada

E-mail address : sudha@accglobal.net