5_tallafha.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 2, No. 2, 2009, (231-238)

ISSN 1307-5543 – www.ejpam.com

Best Approximation in Uniformity type spaces

A. Tallafha∗ and R. Khalil

University of Jordan, Department of mathematics,

Amman-Jordan

Abstract. Let X be a set, and Γ be a collection of subsets of X × X . The object of this paper,

is to define a semi-linear uniform space by assuming certain conditions on Γ. The structure of

such spaces turned to be a very rich structure. We define closest elements from a given set to

a given element in X . Then we study best approximation in semi-linear uniform spaces.

AMS subject classifications: Primary: 41A65, Secondary: 41A99

Key words: Best approximation, Uniform spaces.

1. Introduction

Let X be a set and DX be a collection of subsets of X×X , such that each element V

of DX contains the diagonal ∆= {(x , x) : x ∈ X}and V = V−1 =
�

(y, x) :
�

x , y
�

∈ V
	

for all V ∈ DX (symmetric), DX is called the family of all entourages of the diagonal.

Let Γ be a sub collection of DX , then

The pair (X ,Γ) is called a uniform space if

∗Corresponding author.

Email address: roshdi�ju.edu.jo (R. Khalil)

http://www.ejpam.com 231 c© 2009 EJPAM All rights reserved.



A. Tallafha and R. Khalil / Eur. J. Pure Appl. Math, 2 (2009), (231-238) 232

(i) V1 and V2 are in Γ then V1 ∩ V2 ∈ Γ

(ii) For every V ∈ Γ, there exists U ∈ Γ such that U ◦ U ⊂ V.

(iii) ∩
V ∈ Γ

V =∆

(vi) If V ∈ Γ and V ⊆W ∈ DX , then W ∈ Γ.

Uniform spaces had been studied extensively through years. We refer the reader

to [1], and [2], for the basic structure of uniform spaces. The object of this paper is

to define uniform type spaces and a set valued map, to be called metric type, on such

spaces that enables us to study analytical concepts on uniform type spaces, namely

best approximation. Since the problem of best approximation is a problem of nearness

between elements and sets, the problem of best approximation is usually discussed

in metric and normed spaces [3], [4]. Best approximation never been studied in

spaces other than metric and normed spaces We believe that the new structure that

we introduced in this paper is very fruitful and will give rise to many problems in

approximation theory in uniform spaces.

2. Uniform type spaces

Let (X ,Γ) be a uniform space. By a chain in X × X we mean a totally( or linearly)

ordered collection of subsets of X × X , where V1 ≤ V2 means V1 ⊆ V2.

Definition 1.1. We call (X ,Γ) a semi-linear uniform space if it is a uniform space

where Γ is a chain and condition (vi) is replaced by
⋃

V∈Γ
V = X × X .

An example of a semi-linear uniform space is the following.

Example 2.1. Let Vt = {(x , y) : y − t < x < y + t , and −∞ < y < ∞}. Then

(R,Γ), with Γ = {Vt : 0< t <∞} is a semi-linear uniform space.

One can generate semi-linear uniform spaces as follows. Let DX be a chain in

the power set of X × X , such that, each element of DX is symmetric , contains △,
⋃

U ∈ DX

U = X × X and
⋂

U ∈ DX

U = △. Then one can easily see that (X , DX ) is a semi-



A. Tallafha and R. Khalil / Eur. J. Pure Appl. Math, 2 (2009), (231-238) 233

linear uniform space.

We should remark that the topology in metric and normed spaces can be

generated by semi-linear uniformities.

Throughout the rest of this paper, (X ,Γ) will be assumed semi-linear uniform

space.

Now we introduce one of the main concepts in this paper.

Let (X ,Γ) be a semi-linear uniform space. For x , y ∈ X , let

C (x , y) = ∩{V ∈ Γ : (x , y) ∈ V}, and Σ =
�

C (x , y) : x , y ∈ X
	

.

Clearly C (x , y) = ∩{V−1 ∈ Γ : (x , y) ∈ V}.

Definition 3.1. Let (X ,Γ) be a semi-linear uniform space. We define the set val-

ued map: ρ : X × X → Σ, ρ(x , y) = C (x , y).The map ρ will be called a set metric

on (X ,Γ).

The proof of the following result is immediate and will be omitted.

Proposition 4.1. For a semi-linear uniform space, we have the followings.

(i) ρ(x , y) = ∆ if and only if x = y.

(ii) ρ(x , y) = ρ(y, x).

Now we have the following natural questions.

Question1: Is ρ(x , y)⊆ ρ(x , z)∩ρ(z, y)?.

In metric spaces, it is known that if d(x , y) = d(x , z) then y need not equal z.. In

semi-linear type spaces, the story is different. So we pose the following question.

Question 2. If ρ(x , z) = ρ(x , w),for some x ∈ X . Must w = z ?.

Using the concept of set metric, we introduce the following concepts.

Definition 5.1. For x ∈ X and E ⊂ X , we define ρ(x , E) = ∩
y ∈ E
ρ(x , y).

Clearly, if x ∈ E, then ρ(x , E) = ∆.

Definition 6.1. For x ∈ X and V ∈ Γ, we define The open ball of center x and

radius V to be B(x , V ) = {y : (x , y) ∈ V }. Equivalently B(x , V ) = {y : ρ(x , y) ⊆

V }.Clearly if y ∈ B(x , V ), then there is a W ∈ Γ such that B(y, W )⊆ B(x , V ).



A. Tallafha and R. Khalil / Eur. J. Pure Appl. Math, 2 (2009), (231-238) 234

Definition 7.1. B ⊆ X is called bounded if B ⊆ B(x , V ), for some V ∈ Γ, x ∈ X .

Definition 8.1. Let (xn) be a sequence in X .We say xn converges to x in X , and

we write xn → x , if for every V ∈ Γ there exists k such that (xn, x) ∈ V for every

n ≥ k.

Clearly if xn→ x , then for every j,
∞
⋂

n= j

ρ(x , xn) = ∆.Unfortunately the converse is

not true. But we have

Lemma 9.1. Let (xn) be a sequence in X . If
∞
⋂

n= j

ρ(x , xn) = ∆, for every j, then

there exist a subsequence xnk
→ x .

proof. We may assume that, for every j there is n j ≥ j such that ρ(x , xn j
) 6= ∆,

also we may assume ρ(x , xn j
) is a decreasing sequence and

∞
⋂

j =1

ρ(x , xn j
) = ∆.Let

V ∈ Γ,then there exist j
1

such that ρ(x , xn j
)⊆ V for all j ≥ j

1
, hence xn j

→ x .

Definition 10.1. Let (xn) be a sequence in X , (xn) is called Cauchy if for every

V ∈ Γ there exists k such that (xn, xm) ∈ V for every n, m ≥ k.

Now it is easy to prove the following Corollary

Corollary 11.1. Let (xn) be a Cauchy sequence in X . Then xn → x , iff for every

j ∈ N,
∞
⋂

n = j

ρ(x , xn) = ∆.

Now, we prove:

Lemma 12.1. Let (xn) be a sequence in (X ,Γ). Then.

(i) Every convergent sequence is Cauchy.

(ii) Every Cauchy sequence is bounded.

Proof. (i) Let (xn) converges to x in X , and V ∈ Γ.Let U ∈ Γ such that U ◦ U ⊂ V.

From the definition of convergence, there exists k such that (x , xn) ∈ U for all n > k.

Since U is symmetric, (xm, x) ∈ U for all m> k. Hence (xn, x) ◦ (x , xm) = (xn, xm) ∈

U ◦ U ⊂ V for all n, m > k, and (xn) is Cauchy.

(ii) Let xn be Cauchy, and V ∈ Γ. Then there exists k such that (xn, xm) ∈ V

for every n, m ≥ k.

Let U ∈ Γ be such that
��

xk, x1

�

,
�

xk, x2

�

, ...,
�

xk, xk−1

�	

⊆ U . Then



A. Tallafha and R. Khalil / Eur. J. Pure Appl. Math, 2 (2009), (231-238) 235

�

x1, x2, ...
	

⊆ B
�

xk, W
�

, where W = U ∪V.

Lemma 13.1. Let (xn) be a sequence in X . If (xn) converges then the limit is

unique.

Proof. If possible assume that xn → x and xn → y. Let V ∈ Γ be arbitrary.

Condition (iii) of uniform spaces implies the existence of some W ∈ Γ such that

W ◦W ⊂ V. From the definition of convergence, there exists n◦ such that (x , xn) and

(y, xn) are in W. Hence (x , y) ∈W ◦W ⊂ V. Thus, since V was arbitrary, (x , y) ∈∆,

and so x = y.

Now, a set E will be called open if for every point x in E there exists V ∈ Γ, such

that B(x , V ) ⊆ E. The set E is called closed if E c is open. A point x is called a limit

point of E if there is a sequence (xn) in E such that xn→ x . The set of limit points

of the set E will be denoted by E ℓ. For any set E in X , we let
_

E = E ∪ E ℓ.

The proof of the following lemma is similar to that in metric spaces and will be

omitted.

Lemma 14.1. A set E is closed if and only if E ℓ ⊑ E.

Question 3. If ρ(x , E) = ∆, must x ∈ E ℓ?

Proposition 15.1. If x ∈ E ℓ, then ρ(x , E) = ∆.

Proof. Let x ∈ E ℓ. Then there exists (xn) in E such that xn → x . Hence,
∞
∩

n =1
ρ(x , xn) = ∆. But ρ(x , E) = ∩

y ∈ E
ρ(x , y)⊂

∞
∩

n =1
ρ(x , xn) = ∆. So ρ(x , E) = ∆.

A nice property of semi-linear uniform spaces is:

Theorem 16.1. Open balls separate points in (X ,Γ).

Proof. Let x , y be any two elements in (X ,Γ) such that x 6= y. If possible assume

that B(x , U) ∩ B(y, U) 6= φ for all U ∈ Γ. Let V be any element in Γ. Since X is a

uniform space, then there exists W ∈ Γ such that W ◦W ⊂ V. By assumption B(x , W )∩

B(y, W ) 6= φ. Hence, there exists z ∈ X such that (x , z), (z, y) ∈ W. Consequently,

(x , y) ∈ W ◦W ⊂ V. So (x , y) ∈ V for all V ∈ Γ. But this implies that (x , y) ∈ ∆,

which in turn implies that x = y. This contradicts the assumption. So there must exist



A. Tallafha and R. Khalil / Eur. J. Pure Appl. Math, 2 (2009), (231-238) 236

some W ∈ Γ such that B(x , W )∩ B(y, W ) = φ.

Now, let us define a set E ⊂ (X ,Γ) to be compact, if every sequence in E has a

convergent subsequence in E. Clearly, every finite set is compact, and every compact

set is closed.

3. Proximinality in Semi-Linear Uniform Spaces

What is nice about semi-linear uniform spaces is that theory of best approximation

can be studied in such spaces without tools that metric structure usually offers. In

this section we present some results in approximation theory in semi-linear uniform

spaces.

Definition 1.2. Let (X ,Γ) be semi-linear uniform space, and E ⊂ X . The set E

is called proximinal if for any x ∈ X , there exists some e ∈ E such that ρ(x , E) =

ρ(x , e).

Proposition 2.2. If E ⊂ X is proximinal, then E is closed.

Proof. Let x ∈ E ℓ. By Proposition 11.1, ρ(x , E) = ∆, then by assumption of

proximinality, there exists some e ∈ E such that ρ(x , E) = ρ(x , e) = ∆. So x must

equal e and E is closed.

Compact sets are nice proximinal sets in normed spaces [4]. But what about

proximinality of compact sets in semi-linear spaces.

Question 4. If E is compact, must E be proximinal?.

Every finite set is compact, so the following is a partial answer to our question.

Theorem 2.3. Let (X ,Γ) be a semi-linear uniform space. Then every finite set is

proximinal.

Proof. Since E is finite, then E = {e1, e2, ...en}.Let x ∈ X . Then ρ(x , E ) =
n
∩

i =1
ρ(x , ei). The chain property of semi-linear uniform spaces implies that any two ele-

ments ρ(x , ei), ρ(x , ek) one of them must be contained in the other. Thus {ρ(x , e1), ...ρ(x , en)}



A. Tallafha and R. Khalil / Eur. J. Pure Appl. Math, 2 (2009), (231-238) 237

is a finite chain. Consequently,
n
∩

i =1
ρ(x , ei) = ρ(x , ek) for some k, with 1 ≤ k ≤ n.

Hence E is proximinal.

Corollary 2.4. If E1, E2, ..., En are proximinal in (X ,Γ), then
n
⋃

i =1

Ei is proximinal

too.

Proof. Let x ∈ X . Then ρ(x ,
n
⋃

i =1

Ei) = ∩
y ∈

n
⋃

i=1

Ei

ρ(x , y) =
n
⋂

i =1

( ∩
y ∈Ei

ρ(x , y))). Since Ei

all are proximinal, then ∩
y ∈Ei

ρ(x , y) = ρ(x , ei) for some ei ∈ Ei . Hence ρ(x ,
n
⋃

i =1

Ei) =

n
⋂

i =1

ρ(x , ei) = ρ(x , ek) for some k ∈ {1, 2, ..., n} . So ρ(x ,
n
⋃

i =1

Ei) =
n
⋂

i =1

ρ(x , ei) =

ρ(x , ek) , ek ∈
n
⋃

i =1

Ei .

Also every sequence with it’s limit is compact, so we have another partial answer

to our question.

Theorem 2.5. Let (X ,Γ) be a semi-linear uniform space and (yn) be a convergent

sequence in X . Then E =
¦

y, y
1
, y

2
, ...
©

is proximinal, where y = lim yn.

Proof.Let x ∈ X \ E ( If x ∈ E then ρ(x , E) = ρ(x , x)). So we may assume

ρ(x , yn) 6=∆ for all n. Now if there exist n
0

such that ρ(x , yn0
) ⊆ ρ(x , yn). for all n ,

then ρ(x , E) = ρ(x , yn0
) ∩ρ(x , y) and by Theorem 2.3 we are done. If not, then for

all n there exist mn such that mn < mn+1and ρ(x , ymn
) $ ρ(x , yn) ∩ ρ(x , ymn−1

).So
∞
⋂

n =1

ρ(x , ymn
) =

∞
⋂

n =1

ρ(x , yn).Now we want to show that ρ(x , y) ⊆
∞
⋂

n =1

ρ(x , ymn
).Let

U ∈ Γ be such that (x , ymn
) ∈ U , for some mn ∈ N, therefor (x , ymk

) ∈ U for all j ≥ n.

let Wj ∈ Γ be such that B
�

x , 2Wj

�

× B
�

ymk
, 2Wj

�

⊆ U , also we may assume that

Wj ⊇ Wj+1.If
∞
⋂

j =1

Wj = ∆,let
�

t j, s j

�

∈ B
�

x , Wj

�

× B
�

ymk
, Wj

�

, then t j → x and s j

→ y (lim s j = lim ymk
).



A. Tallafha and R. Khalil / Eur. J. Pure Appl. Math, 2 (2009), (231-238) 238

therefor
�

x , y
�

∈
−−−−−−−−

B
�

x , Wj

�

×
−−−−−−−−−−

B
�

ymk
, Wj

�

⊆ B
�

x , 2Wj

�

× B
�

ymk
, 2Wj

�

⊆ U .

Now if
∞
⋂

j =1

Wj 6= ∆, then there exist W ∈ Γ such that W ⊆
∞
⋂

j =1

Wj, so B (x , W)×

B
�

ymk
, W
�

⊆ U for all j.Since ymk
−→ y ,

there exist N such that
�

ymN
, y
�

∈W , so
�

x , y
�

∈ U , and the result follows.

References

[1] Engelking, R. Outline of General Topology, North-Holand, Amsterdam, 1968.

[2] James, I.M. Topological and Uniform Spaces. Undergraduate Texts in Mathe-

matics. Springer-Verlag 1987.

[3] Light, W. and Cheney, E. Approximation theory in tensor product spaces. Lec-

ture Notes in Math. 1169. Springer Verlag, New York, 1985.

[4] Singer, I. Best approximation in normed linear spaces by elements of linear

subspaces. Springer-Verlag, New York, 1970.