YangAGT.dvi


@
Applied General Topology

c© Universidad Politécnica de Valencia

Volume 10, No. 1, 2009

pp. 49-68

Pointwise convergence and Ascoli theorems for
nearness spaces

Zhanbo Yang
∗

Abstract. We first study subspaces and product spaces in the

context of nearness spaces and prove that U-N spaces, C-N spaces, P-

N spaces and totally bounded nearness spaces are nearness hereditary;

T-N spaces and compact nearness spaces are N -closed hereditary. We

prove that N2 plus compact implies N -closed subsets. We prove that

totally bounded, compact and N2 are productive. We generalize the

concepts of neighborhood systems into the nearness spaces and prove

that the nearness neighborhood systems are consistent with existing

concepts of neighborhood systems in topological spaces, uniform spaces

and proximity spaces respectively when considered in the respective

sub-categories. We prove that a net of functions is convergent under the

pointwise convergent nearness structure if and only if its cross-section

at each point is convergent. We have also proved two Ascoli-Arzelà

type of theorems.

2000 AMS Classification: 54E17, 54E05, 54E15, 68U10

Keywords: Nearness spaces, subspace, product space, neighborhood system,
pointwise convergent, Ascoli’s theorem

1. Introduction

As a natural extension of geometry, the concept of “near/apart” has been a
center for topology and related studies. Topology characteries the “nearness”
between a point and a set. Proximity [14] is an axiomatization of “nearness”
between two sets. Contiguity [10] describes the concept of nearness among the
elements of a finite family of sets. The concept of “nearness space” introduced
by Herrlich [8] in 1974 attempts to characterize the nearness of an arbitrary
collection of sets. The category of nearness spaces, the most general among
the aforementioned structures, can be used as a unifying framework. The

∗This work was in part supported by a grant from the 2008 Faculty Research Fund of the
University of the Incarnate Word.



50 Z. Yang

categories of several aforementioned structures can all be “nicely embedded”
into the category of the nearness spaces as (either bireflective or bicoreflective)
sub-categories ([8]).

In recent years, the notion of “nearness” in a number of different variations
has found new applications in digital topology, image processing and pattern
recognition areas, perhaps due to the fact that those structures are “richer”
than classical topology. In 1995, Latecki and Prokop [12] used a weaker ver-
sion of proximity spaces called semi-proximity spaces (sp-spaces). They talked
about the possibility of describing all digital pictures used in computer vision
and computer graphs as non-trivial semi-proximity spaces, which is not possible
in classical topology. They also discussed the application of “semi-proximity
continuous functions” in well-behaved operations such as thinning on digital
images. In 1996, Chaudhuri [4] introduced a new definition for the neighbors
of an arbitrary point P . This new definition used a “centroid criterion” to
capture the idea that the neighbors of P should be as near to P and as sym-
metrically paced around P as possible. This new definition could be used for
pattern classification, clustering and low-level description of dot patterns. In
2000, Ptak and Kropatsch [16] discussed the application of proximity spaces
in studying of digital images. They showed by examples that the “proximate
complexity” of a finite covering in a digital picture might be too high to be
adequately depicted in a finite topological space, which might indicate another
conceptual advantage of proximities over topologies. Most recently, in 2007,
Wolski and Peters [18, 15] investigated approximation spaces in the context
of topological structures which axiomatized certain notion of nearness. Peters
[15] pointed out particularly that the concept of “nearness” was not confined to
spatial nearness, or geometrical likeness. It was possible to introduce a nearness
relation that could be used to determine the “nearness” among sets of objects
that were spatially far away and, yet, “qualitatively” near to each other.

The main objectives of this paper are to establish a ”pointwise convergent”
nearness structure on a function space made of a family of functions from X
to Y and to establish two versions of the Ascoli-Arzelà theorems for nearness
spaces that relate the compactness of the underlining space Y with that of the
function space. Since the function space is really a subspace of the product
space Y X , we begin with the nearness structures on a subspace and discuss the
hereditary properties for a number of important concepts in nearness spaces.
We then define the nearness structure on a product spaces and discuss its var-
ious properties. The nearness structure on a function space is then introduced
as a subspace of the product space Y X . We will end the discussion with two
Ascoli-Arzelà type of theorems.

Some work in the past, such as [2] (1979), [7] (1979) and [1] (2006) have
discussed a number of results with respect to subspaces and product spaces.
Most of the results in that paper were dealing with topological nearness (T-N )
spaces and do not duplicate what are to be presented in this paper. For the
purpose of clarity and being self-contained, we will still give the definition of
”subspace” and ”product” space here and prove relevant results.



Pointwise convergence for nearness spaces 51

The classical Ascoli-Arzelà theorem was proved in the 19th century first
by Ascoli and then independently by Arzelà. It characterizes compactness of
sets of continuous real-valued functions on the interval [0,1] with respect to
the topology of uniform convergence. It is commonly known that the issue
came from the fact that a convergent sequence of continuous functions may
not converge to a continuous function. So the natural question is: under what
conditions the limit of a convergent sequence of continuous functions is still
continuous. It turned out that the concept of equicontinuous was used to
characterize the condition needed (see [11]) in topological spaces. In 1970,
[13] discussed Ascoli’s theorem for the spaces of multifunctions. In 1981, [6]
discussed Ascoli’s theorem for topological categories. In 1984, [3] discussed
Ascoli’s theorem for a class of merotopic spaces. In 1993, [5] studied a version
of the Ascoli’s theorem for set valued proximally continuous functions. In 2001,
[17] proved a version of Ascoli’s theorem for sequential spaces. As far as we
know, no nearness space version of the Ascoli’s theorem has been established
yet at this time.

We have practical reason to be interested in this topic. In many cases, a
digital image processing algorithm is essentially the application of a sequence
of deformation functions to a digital plane. For example, [12] proved that a
deletion of a simple point (a point that does not affect the connectness of the
digital picture) can be regarded as a sp-continuous function. Hence a thinning
algorithm that preserves connectness can be arranged as a sequence of sp-
continuous functions. We may be able to use the tools of function spaces, and
the results on convergence of function sequences to study the image processing
algorithms, which opens a new set of doors.

The rest of this paper is organized as follows: Section 2 is a collection of
the major definitions involving nearness spaces that are relevant to this paper.
Section 3 studies the nearness structures on a subspaces. Section 4 is about
the product spaces and the function spaces. The summary at the end concludes
this paper.

2. Notation and Definitions

In this section, we define the basic concepts used throughout this paper.
We will use the language of Categories in some of our discussions. For

readers who are not familiar with Category theory, a category is basically a
family of objects with a particular type of structures. For example, we can
talk about the category of all topological spaces, the category of all groups,
etc. The so called “morphism” from one object to another is a function that
preserves the structure on the objects. For example, a “morphism” in the
category of topological spaces would be a continuous function. A ”morphism”
in the category of all groups would be a homomorphism. An embedding from
one category into another category is a way to assign each object from one
category to an object of the other category in some injective manner that also
preserves the morphisms. For readers who are interested at further information
about category theory, please see [20].



52 Z. Yang

The readers can see Kelley [11] or any common general topology text book
for terms in general topology.

2.1. Basic Notations. Let X be a set, P(X) represents the power set of X.
P0(X) = X, P1(X) = P(X), . . . , Pn(X) = P(Pn−1(X))

A, B, . . . represent elements in P(X), i.e. subsets in X
A , B, . . . represent elements in P2(X), i.e. subsets in P(X)
ξ, η, . . . represent elements in P3(X), i.e. subsets in P2(X)
A

C = {X − A : A ∈ A }
For each B ⊆ X, A (B) =

⋃

{A : A ∩ B 6= φ, A ∈ A }
ξA denotes A ∈ ξ, ξA denotes A /∈ ξ
Aξ B denotes {A, B} ∈ ξ. Aξ B denotes {A, B} /∈ ξ
clξA = {x : {x} ξ A}, intξA = X − clξ(X − A)
clξA = {clξA : A ∈ A }, intξA = {intξA : A ∈ A }
A ∨ B = {A ∪ B : A ∈ A , B ∈ B}
A ∧ B = {A ∩ B : A ∈ A , B ∈ B}
A ≺ B if and only if for any B ∈ B , there is A ∈ A such that A ⊆ B.

This is referred to as ”B co-refines A ”.

2.2. Definitions Related to Nearness Structure. We will restate some
major definitions about nearness spaces here (due to [8]):

(i) Let X be a non-empty set. The ordered pair (X, ξ) is said to be a
nearness space, or N -space, if the following are satisfied:

(N1) If
⋂

A 6= φ, then ξA .
(N2) If ξB, and for each A ∈ A , there exists a B ∈ B such that

B ⊆ clξA, then ξA , i.e. B ≺ clξA .

(N3) If ξA and ξB, then ξ(A ∨ B).
(N4) If φ ∈ A, then ξA .

(ii) Let X be a set. Let (X, ξ) and (Y, η) be two N-spaces. A function
f : X → Y is said to be a (ξ, η) N-preserving map, or an N-preserving
map, or simply an N-mapping, if one of the following two equivalent
conditions is satisfied:

(M1) If ξA , then η f (A ), where f (A ) = {f (A) : A ∈ A }

(M2) If ηB, then ξ f −1(B), where f −1(B) = {f −1(B) : B ∈ B}.
We will use the notation NEAR to represent the category of all

nearness spaces with N-mappings.
(iii) An N-space is called an N1-space, if the following is satisfied:

(N0) If {x}ξ{y}, then x = y.
(iv) An N-space is called an N2 space, if for any x, y ∈ X, x 6= y implies the

existence of A ⊆ X and B ⊆ X such that A ∩ B = φ, ξ{{x}, X − A}
and ξ{{y}, X − B}.

(v) An N-space is called a T-N space, if the following is satisfied:
(T) If ξA , then

⋂

clξA 6= φ.
A T − N2 space is an N2 space that also satisfies the condition (T).



Pointwise convergence for nearness spaces 53

We will use the symbol T − NEAR to represent the subcategory of
NEAR, consists of all T-N spaces with N-mappings.

(vi) An N-space is called a U-N space, if the following is satisfied:

(U) If ξA , then there exists a B such that ξB and for each B ∈ B,
there is an A ∈ A such that BC(X − B) ⊆ X − A.

We will use the notation U − NEAR to represent the subcategory of
NEAR, consists of all U-N spaces with N-mappings.

(vii) An N-space is called a C-N space, if the following is satisfied:
(C) If ξA , then there is a finite subcollection B ⊆ A such that ξB.
We will use the notation C − NEAR to represent the subcategory of

NEAR, consists of all C-N spaces with N-mappings.
(viii) An N-space is called a P-N space, if it satisfies both of the conditions

(U) and (C).
We will use the notation P − NEAR to represent the subcategory of

NEAR, consists of all P-N spaces with N-mappings.
(ix) An N-space is called a totally bounded space, if one of the following

equivalent conditions is satisfied:
(B1) If ξA , then there is a finite subcollection B ⊆ A such that

⋂

B = φ.
(B2) If F is a filter on X, then ξF .

(x) An N-space is called a compact space, if it satisfies both condition (T)
and (C).

(xi) Let X be a set. {ξα : α ∈ Λ} is a family of N-structures on X. The least
upper bound, denoted by ξ = sup{ξα : α ∈ Λ}, is defined as follows:

ξA if and only if there are finitely many Ai’s such that for each

i = 1, 2, ...n, ξiAi, and A ≺
n
∨

i=1
Ai.

(xii) Let (X, ξ) be a nearness space. Then the topology induced by the
closure operator A 7→ clηA is denoted by Tξ.

3. Nearness Structure on Subspaces

We will begin by giving the definition of a ”nearness subspace”, then proceed
to show that subspaces as defined here are well-defined (Theorem 3.2), act
”natural” (Lemma 3.3) and produces a topology that is consistent with the
subspace topology (Theorem 3.8).

Definition 3.1. Let (X, ξ) be a nearness space and X0 ⊆ X. Define ξ0 on X0
as follows:

ξ0 = {A ⊆ P(X0) : ξA }.

We will denote such ξ0 as ξ0 = ξ|X0 and refer to it as the “nearness structure
on the subspace induced by the nearness structure ξ”.

Theorem 3.2. Let (X, ξ) be a nearness space, and X0 ⊆ X. ξ0, as defined in
Definition 3.1, is a nearness structure on X0.



54 Z. Yang

The proof is an easy deduction from the fact that ξ0 is consists of the type
of A that are in ξ already. We will skip the details. The following lemma is
also easy to prove.

Lemma 3.3. Let (X, ξ) be a nearness space and X0 ⊆ X. Let i : X0 → X be
the inclusion map, then ξ0 = i

−1(ξ). Hence i : X0 → X is N-preserving.

Lemma 3.4. Let X be a set, (Y, η) be a nearness space. Let f : X → Y be an
injective map. Then

Tf −1(η) = f
−1(Tη).

Proof.

clf −1(η)A = {x : {x}f
−1(η)A}

= {x : f (x)η f (A)}

= {x : f (x) ∈ clη(f (A))}

= {x : x ∈ f −1(clη(f (A))}

= f −1(clη(f (A)).

�

We will next exam whether some of the common properties are hereditary.
i.e. whether a particular condition or property can be “inherited” by its sub-
spaces from their “mother” spaces. It turns out that being a T-N space is not
hereditary (Example 3.5). Neither was being a compact N-space. Those two
properties can be inhered by N-closed subspaces. Many of the other properties
are hereditary.

Example 3.5. A subspace of a T-N space may not be a T-N space.

Let X = R, the real line with an ordinary open interval topology T . Then
(X, T ) is a R0- space, hence corresponding to a T-N space (X, ξ) ([8] Theorem
2.2). Now we let X0 = X − {0}, A = (−∞, 0) and B = (0, ∞). Then clξA ∩
clξB 6= φ, but clξ0 A ∩ clξ0 B = φ. This means that the nearness subspace
(X0, ξ0) does not satisfy condition (T), hence not an T-N space.

Definition 3.6. Let (X, ξ) be a nearness space and (X0, ξ0) be a subspace.
(X0, ξ0) is said to be a N-closed subspace, if for any A ⊆ X0, we have clξ0 A =
clξA.

Theorem 3.7. Let (X, ξ) be a nearness space and (X0, ξ0) be a subspace. Then
(X0, ξ0) is a N-closed subspace if and only if clξ0 X0 = clξX0.

Proof. The necessity is obvious. We now will prove the sufficiency. Take any
A ⊆ X0, then

clξ0 A = clξA ∩ X0

= clξA ∩ clξX0

= clξ(A ∩ X0)

= clξA.



Pointwise convergence for nearness spaces 55

�

Theorem 3.8. Let (X, ξ) be a nearness space and (X0, ξ0) be a subspace. Then
(a) Tξ0 = Tξ|X0 .

(b) If (X, ξ) is a T-N space and (X0, ξ0) is an N-closed subspace, then (X0, ξ0)
is also a T-N space.

Proof. (a) Tξ0 = Ti−1(ξ) = i
−1(Tξ) = Tξ

∣

∣

X0
.

(b) Let ξ0A0, then ξA0. It follows from the assumption of (X0, ξ0) being an
N-closed subspace that clξA0 = clξ0 A0. Therefore

⋂

{clξ0 A0 : A0 ∈ A0} =
⋂

{clξA0 : A0 ∈ A0} 6= φ. So (X, ξ0) satisfies the condition (T). �

Theorem 3.9. Let (X, ξ) be a nearness space and (X0, ξ0) be a subspace.
Then if (X, ξ) is a U-N space, so is (X0, ξ0). Moreover, Uξ0 = Uξ|X0 , where
Uξ0 and Uξ denotes the uniformity induced by the nearness structure ξ0 and ξ
respectively. Uξ|X0 is the uniformity Uξ restricted to X0.

Proof. If ξ0A0, then ξA0. Since (X, ξ) is an U-N space, there exists a ξB that
satisfies the condition (U) with respect to A0. Let B0 = {B ∩ X0 : B ∈ B}.

From (N2) we can see that ξ0B0. For each B0 ∈ B0 = {B ∩ X0 : B ∈ B},
there is a B ∈ B such that B = B ∩ X0. So by condition (U), there should
be an A0 ∈ A0 such that A0 ⊆

⋂

{C : B ∪ C 6= X, C ∈ B}. Also because
A0 ⊆ X0, we have

A0 = A0 ∩ X0

⊆
⋂

{C : B ∪ C 6= X, C ∈ B} ∩ X0

=
⋂

{C ∩ X0 : B ∪ C 6= X, C ∈ B}

⊆
⋂

{C0 : B0 ∪ C0 6= X0, C0 ∈ B0}

Therefore, ξ0 satisfies the condition (U). Furthermore, for any A0 ∈ P
2(X), we

have the following equivalent deductions:

A0 ∈ U |i−1(ξ)

⇔ A C0 /∈ i
−1(ξ)

⇔ A C0 /∈ ξ0

⇔ A0 ∈ U |ξ0

⇔ A0 ∈ i
−1(Uξ).

Therefore,

Uξ0 = Ui−1(ξ) = i
−1(Uξ) = Uξ|X0 .

�



56 Z. Yang

Theorem 3.10. Let (X, ξ) be a nearness space and (X0, ξ0) be a subspace. If
(X, ξ) is a C-N space, so is (X0, ξ0). Moreover, Cξ0 = Cξ|X0 , where Cξ0 and Cξ
denote the contiguity induced by the nearness structure ξ0 and ξ respectively.
Cξ|X0 is the contiguity Cξ restricted to X0.

Proof. For any A0 ∈ P
2(X) and ξ0A0, then ξA0. By condition (C), A0 has a

finite subcollection B0 such that ξB0. This implies that ξ0B0. And further-
more,

A0 ∈ Cξ0

⇔ A0 ∈ ξ0 and A0 is finite.

⇔ A0 ∈ ξ, A0 is finite

⇔ A0 ∈ Cξ

⇔ A0 ∈ Cξ|X0 .

�

Since a P-N space is one that satisfies condition (U) and (C), the following
Theorem is obvious from the Theorems 3.9 and 3.10:

Theorem 3.11. Let (X, ξ) be a nearness space and (X0, ξ0) be a subspace.
Then if (X, ξ) is a P-N space, so is (X0, ξ0).

Since a compact nearness space is one that satisfies condition (T) and (C),
the following theorem is obvious from Theorems 3.8 and 3.10:

Theorem 3.12. Let (X, ξ) be a nearness space and (X0, ξ0) be a N-closed
subspace. Then if (X, ξ) is a compact nearness space, so is (X0, ξ0).

Lemma 3.13. Let (X, ξ) be a T-N space. Then (X, clξ) is topologically compact
if and only if (X, ξ) is a compact nearness space.

Proof. Recall that (X, ξ) is a compact nearness space if and only if condition
(T) and (C) are satisfied.

Necessity: If (X, clξ) is topologically compact. Take an A such that ξA .
We want to show that condition (C) is met by showing that A has a finite

subcollection B such that ξB. We will first claim that
⋂

clξA = φ. If not,
then by (N1), ξclξA would be true. For each A ∈ A , there would be a
clξA ∈ clξA such that clξA ⊆ clξA. By (N2), ξA would be true, and that

would contradict to the assumption of ξA . So
⋂

clξA = φ must be true.
(clξA )

C is an open cover of X. Since (X, clξ) is topologically compact, we
will let B be the finite subcollection of A and (clξB)

C is an open cover of X.

This implies that
⋂

clξB = φ. By condition (T), ξB is true. Hence condition
(C) has been met.
Sufficiency: If (X, ξ) is a compact nearness space, which means that it
meets condition (T) and (C). Any open cover of (X, clξ) can be expressed as
the complement collection of a collection clξA and

⋂

clξA = φ. From

condition (T), ξclξA is true. From condition (C), there must be a finite



Pointwise convergence for nearness spaces 57

subcollection of clξA , say clξB, such that ξclξB is true. It follows that
⋂

clξB = φ. Then (clξB)
C is the finite subcover of the original open cover.

Hence (X, clξ) is topologically compact.

�

From Lemma 3.13, one can easily see the following is true:

Lemma 3.14. Let (X, ξ) be a compact nearness space, (Y, η) be a T-N space
and f : (X, ξ) → (Y, η) be N-preserving, then (Y, η) is a compact nearness
space.

Definition 3.15. Let X be a set. Define a partial order among all possible
nearness structures on X as follows: If ξ1 and ξ2 are two nearness structures
on a set X, then ξ1 ≤ ξ2 if and only if ξ1 ⊇ ξ2.

Lemma 3.16. (i) Let X be a set and ξ1 and ξ2 be two nearness structures
on a set X. ξ2 ≤ ξ1 if and only if i : (X, ξ1) → (X, ξ2) is N-preserving.

(ii) Let (X, ξ) and (Y, η) be two nearness spaces. Then f : (X, ξ) → (Y, η)
is N-preserving if and only if ξ ≥ f −1(η).

Proof. (i)

ξ2 ≤ ξ1

⇔ ξ2 ⊇ ξ1

⇔ i(ξ1) ⊆ ξ2

⇔ i : (X, ξ1) → (X, ξ2) is N − preserving.

(ii)

f : (X, ξ) → (Y, η) is N − preserving

⇔ ξ ⊆ f −1(η)

⇔ ξ ≥ f −1(η).

�

Theorem 3.17. Let (X, ξ) be a T − N2 space and (X0, ξ0) be a N-compact
subspace. Then (X0, ξ0) is N-closed.

Theorem 3.18. If (X, ξ) is totally bounded, and X0 ⊆ X, then (X0, ξ0) is
totally bounded.

Proof. ξ0A0 implies that ξA0. Hence, by condition (B1) for (X, ξ), there is
a B0, a finite subcollection of A0, such that

⋂

B0 = φ. So (X0, ξ0) satisfies
(B1). �

The following lemma proved by Hunsaker and Sharma as Corollary (2.5) in
their 1974 paper [9] is used to prove the next theorem.

Lemma 3.19. Let f : (X, ξα) → (Y, ηα) be an N-preserving map for each
α ∈ Λ. Then f : (X, sup{ξα}) → (Y, sup{ηα}) is an N-preserving map.



58 Z. Yang

The following theorem is needed to ensure that the concept of nearness
structure on the function space, which will be introduced in next section, is
well defined.

Theorem 3.20. If {ξα : α ∈ Λ} is a family of nearness structures on X and
X0 ⊆ X. Then

sup
α∈Λ

{ξα}

∣

∣

∣

∣

X0

= sup
α∈Λ

{ ξα|X0}.

Proof. Let i : X0 → X be the inclusion map. For each α ∈ Λ, By Lemma 3.3,

i : (X0, ξα|X0 ) → (X, ξα)

is N-preserving. From Lemma 3.19,

i : (X0, sup
α∈Λ

{ξα

∣

∣

∣

∣

X0

}) → (X, sup
α∈Λ

{ξα})

is still N-preserving. By Lemma 3.16,

sup
α∈Λ

{ξα

∣

∣

∣

∣

X0

} ≥ i−1(sup
α∈Λ

{ξα}) = sup
α∈Λ

{ξα}

∣

∣

∣

∣

X0

.

Moreover, for each α ∈ Λ,

sup
α∈Λ

{ξα} ⊆ ξα,

hence

sup
α∈Λ

{ξα}

∣

∣

∣

∣

X0

⊆ ξα|X0 .

It follows that

sup
α∈Λ

{ξα}

∣

∣

∣

∣

X0

⊆ sup
α∈Λ

{ξα

∣

∣

∣

∣

X0

}.

i.e.

sup
α∈Λ

{ξα}

∣

∣

∣

∣

X0

≥ sup
α∈Λ

{ ξα|X0}.

Therefore,

sup
α∈Λ

{ξα}

∣

∣

∣

∣

X0

= sup
α∈Λ

{ ξα|X0}.

�

4. The Pointwise Convergent Nearness Structure on Function
Space

The following theorem shows that the least upperbound nearness structure is
a generalization of the respective least upperbound structure when considered
in each of the subcategory of T − NEAR, U − NEAR and P − NEAR respec-
tively.



Pointwise convergence for nearness spaces 59

Figure 1. Commutative diagram of the natural projections

Theorem 4.1. If {ξα : α ∈ Λ} is a family of nearness structures on X and let
ξ = sup

α∈Λ
{ξα}. Then, when considered in T − NEAR, U − NEAR, or P − NEAR,

ξ will induce the least upper bound of the respective structures induced by {ξα :
α ∈ Λ} in the respective type of spaces.

Proof. Let F be the isomorphic functor from T − NEAR to R0 − TOP. We just
need to prove that F is order preserving. Assume ξ1 ≤ ξ2. By Lemma 3.16,
ξ1 ≤ ξ2 ⇔ i : (X, ξ2) → (X, ξ1) is N-preserving ⇔ F (i) is a morphism from
F [(X, ξ2)] to F [(X, ξ1)]1) ⇔ Tξ2 ⊇ Tξ1 . This shows that F does preserve the
order.
The proofs for other two types are parallel and therefore omitted.

�

The following theorem ensures that the product nearness structure in the
categorical sense is the largest nearness structure on the product space that
makes all natural projections N-preserving.

Theorem 4.2. If {(Xα, ξα) : α ∈ Λ} is a family of nearness spaces and let Pα :
∏

α∈Λ Xα → Xα be the natural projection map. Then the nearness structure

ξ∗ = sup
α∈Λ

{P −1α (ξα)} is exactly the categorical product of {(Xα, ξα) : α ∈ Λ}.

Proof. It would suffice to show that for any N-space (X, ξ), and any family of
N-preserving maps {fα : X → Xα : α ∈ Λ}, there is an unique N-preserving
map f : X →

∏

α∈Λ Xα such that ∀α ∈ Λ, Pα ◦ f = fα. i.e. the diagram in
Figure 1 is commutative.

We will first make a claim that for any map f : X →
∏

Xα, f is N-preserving
if and only if for each α ∈ Λ, Pα ◦ f is N-preserving. In fact, the necessity is
obvious. Let us assume that for each α ∈ Λ, Pα ◦ f is N-preserving. From
Lemma 3.19, f is (ξ, sup

α∈Λ
{P −1α }) N-preserving. i.e. it is (ξ, ξ

∗) N-preserving.

Now we consider the family of N-preserving maps {fα : X → Xα : α ∈ Λ}.
Define f in the natural way (usually known as the “evaluation map”): ∀x ∈
X, f (x) = (fα(x))α∈Λ. Then Pα ◦ f = fα. Since each fα is N-perserving, each
Pα ◦ f is N-preserving. By earlier proof, f is N-preserving. We also know that
such an f is unique from its definition. �



60 Z. Yang

The following purely categorical lemma should be obvious:

Lemma 4.3. If C is a category, {Aα : α ∈ Λ} is a family of objects.
∏

α∈Λ Aα
is the categorical product in C. D is another category isomorphic to C with F
as the isomorphic functor from C to D. Then F (

∏

α∈Λ Aα) =
∏

α∈Λ F (Aα).

We now officially define the nearness structure on the function space:

Definition 4.4. If (Y, ξ) is a N-space, Xis a non-empty set. F ⊆ Y X . If
we consider Y X as a product and let ξ∗ be the product nearness structure as
defined in Theorem 4.2. Let ξρ = ξ

∗|
F

. Then ξρ is said to be the pointwise
convergent nearness structure on F ⊆ Y X .

Notice that if {ex : Y
X → Y : x ∈ X} is the family of natural projections,

then ξ∗ = sup
x∈X

{e−1x (ξ)}. So by Theorem 3.20, ξρ = ξ
∗|

F
= sup

x∈X

{e−1x (ξ)}

∣

∣

∣

∣

F

=

sup
x∈X

{ e−1x (ξ)
∣

∣

F
}.

The readers may refer to [11] for the concept of product topology, product
uniformity, pointwise convergent topology and pointwise convergent uniformity.
Refer to [14] for the concepts of product proximity and pointwise convergent
proximity.

We would like to make sure that the product nearness structure and the
pointwise convergent nearness structure is a generalization of the respective
structures in topological spaces, uniform spaces and proximity spaces respec-
tively.

Theorem 4.5. When considered in each of the subcategory T − NEAR,
U − NEAR, or P − NEAR,

(i) ξ∗ will induce the Tychonoff product topology, product uniformity or
the product proximity respectively.

(ii) ξρ will induce the pointwise convergent topology, the pointwise conver-
gent uniformity or the pointwise convergent proximity respectively

Proof. The first conclusion can be obtained from Theorems 4.1 and 4.2. The
second conclusion can be obtained from Theorems 3.8, 3.9 and 3.10. �

Next we will try to generalize the concept of ”neighborhood”, which is es-
sential when characterizing ”convergence”.

Definition 4.6. If (X, ξ) is a N-space, and A ⊆ X. A subset U is called a
nearness neighborhood of A if there is a ξA such that A ⊆ A C(A) ⊆ U . The
notation N earN (A) represents the collection of all nearness neighborhood of a
subset A. If the set A contains only one point x, then we simplify the notation
from N earN ({x}) to N earN (x).

If there are two types of neighborhood system on a space that characterize
the same convergence, i.e. being convergent under one neighborhood system
is equivalent to being convergent under the other neighborhood system, then
we consider them as ”equivalent” neighborhood systems. This is typically



Pointwise convergence for nearness spaces 61

characterized by the condition that any neighborhood under one system always
contains a neighborhood in the other system. For example, consider the two
dimensional Cartesian plane. The neighborhood of circular disks centered at a
point is equivalent to the neighborhood of squares centered at the same point.

We will try to show that the nearness neighborhood generalizes the topo-
logical neighborhood([11]), uniform neighborhood ([11]), and proximal neigh-
borhood ([14]) by showing that the nearness neighborhood system is really
equivalent to the respective neighborhood system in the respective subcate-
gories.

Theorem 4.7. Let (X, ξ) be a N-space, x ∈ X and A ⊆ X.

(i) In T − NEAR, N earN (x) is equivalent to a topological neighborhood
system at x.

(ii) In U − NEAR, N earN (x) is equivalent to a uniform neighborhood sys-
tem at x.

(iii) In P − NEAR, N earN (A) is equivalent to a proximal neighborhood sys-
tem at A.

Proof. (i) Take an U ∈ N earN (x), there will be an A /∈ ξ such that
x ∈ A C(x) ⊆ U . For this A , by conditions (N1) and (N2),

⋂

clξA =
∅. So (clξA )

C is a cover of X. There will be an A ∈ A such that
x0 ∈ X − clξA ⊆ X − A ⊆ A

C(x0). The set X − clξA is an open
neighborhood of x.

On the other hand, if we take an open neighborhood V of x under
the topology Tξ, then x /∈ X − V and X − V is closed, i.e. clξ(X −
V ) = X − V . Let A = {{x}, X − V }, then A /∈ ξ. And A C(x) =
X − (X − V ) = V ⊆ V . This shows that V is a nearness neighborhood
also.

(ii) Take an U ∈ N earN (x), there will be an A /∈ ξ such that x ∈ A C(x) ⊆
U . For this A , since A /∈ ξ, A C is a cover of X, which means that
there has to be an A ∈ A such that x ∈ X − A. Let

UA = {
⋃

A∈A

((X − A) × (X − A)) : A /∈ ξ},

then

UA [x] = {y : (x, y) ∈ UA }

= {y : ∃A ∈ A ∋ (x, y) ∈ (X − A) × (X − A)}

= {y : ∃A ∈ A ∋ x ∈ (X − A), y ∈ (X − A)}

=
⋃

{X − A : x ∈ X − A}

= A C(x)

This “equal” relation shows the equivalency between the uniform neigh-
borhood system and the nearness neighborhood system.

(iii) [14] stated that a set B is a proximal ( δ−) neighborhood of a set A if
A ≪ B. In [19], the proximity ≪ξ induced by a nearness structure is



62 Z. Yang

defined as A ≪ξ B if and only if there is a A /∈ ξ such that A
C(A) ⊆ B.

So the equivalency between the nearness neighborhood system and the
proximal neighborhood systems is obvious.

�

The following theorem demonstrates the consistency of the N -converges with
those previously established concepts of convergent nets. With the establish-
ment of the previous theorem, the proof should be obvious.

Theorem 4.8. The N-convergence, as it is defined in Definition 4.9, when
considered in T − NEAR, U − NEAR and P − NEAR, is equivalent to the con-
vergence with respect to the corresponding types of structures, respectively.

A set D is said to be a directed set, if it is endowed with a reflexive and
transitive binary relation ≥ such that ∀m, n ∈ D, ∃p ∈ D s.t. p ≥ m and p ≥ n.
i.e. for any two elements of D, there is always another element that precedes
them. (see [11])

As a generalization of sequences, a net in a set X is a function x : D → X,
where D is a directed set. We typically write a net as {xd : d ∈ D}.

We would like to exam the relation of the convergency of a net of functions
and that of the nets obtained by fixing the net of functions at any arbitrary
point x of X. Of course, we expect the two convergences are to be equivalent.
Theorem 4.7 shows exactly that.

Definition 4.9. If (X, ξ) is a N-space, {xd : d ∈ D} is a net in X. We say
{xn : n ∈ D} N-converges to a point x0 ∈ X, if for any ξA , there is an N ∈ D
such that for each n ≥ N, n ∈ D, we have xn ∈ A

C(x0).

Theorem 4.10. If (Y, ξ) is a N-space, Xis a non-empty set. {fn : n ∈ D} is
a net in F ⊆ Y X . Then {fn : n ∈ D} N-converges to a function f in (F , ξρ)
if and only if for any x ∈ X, {fn(x) : n ∈ D}, as a net in (Y, ξ), N-converges
to the point f (x).

Proof. Necessity: Take an arbitrary point x ∈ X, take an ξA , since the
natural projection map

ex : Y
X → X

is N-preserving, we have ξρe
−1
x (A ), so ξρ( e

−1
x (A )

∣

∣

F
). Since {fn : n ∈ D} is

N-convergent to f in F . There is an n ∈ D, such that for each
m > n, m ∈ D, we have

fm ∈ ( e
−1
x (A )

∣

∣

F
)C (f ).

i.e. There is an A ∈ A , such that

{fm, f} ⊆ F − e
−1
x (A).

So
fm(x) = ex(fm) /∈ A

and
f (x) = ex(f ) /∈ A.



Pointwise convergence for nearness spaces 63

i.e.

fm(x) ∈ X − A,

and

f (x) ∈ X − A

Therefore,

fm(x) ∈ A
C (f (x)).

Sufficiency: Assume that {fn : n ∈ D} is a net in F ⊆ Y
X . Furthermore,

for any x ∈ X, assume that {fn(x) : n ∈ D}, as a net in X, N-converges to
the point f (x). We now arbitrarily take a B such that ξρB. By the definition
of ξ∗ as the least upper bound, there should be finitely many
Bi ⊆ P(X), i = 1, 2, ..., n, such that for each i, we have Bi /∈ e

−1
x (ξ)

∣

∣

F
, and

B ≺ ∨Bi. ξexi (Bi), so there is an mi such that for any n ≥ mi, there should
be an Ai ∈ exi (Bi), {fn(xi), f (xi)} ⊆ X − Ai. Since Ai ∈ exi (Bi), there
exists a Bi ∈ Bi and Ai = exi (Bi). So fn(xi) /∈ exi (Bi), and f (xi) /∈ exi (Bi).
i.e. exi (fn) /∈ exi (Bi), and exi (f ) /∈ exi (Bi). Therefore, {fn, f} ⊆ F − Bi, or
we can say that fn ∈ B

C
i (f ). Since there are only finitely many mi’s. We will

let N = max{m1, ..., mn}. Then for any n ≥ N ,
fn ∈ ∧(B

C
i )(f ) = (∨Bi)

C (f ). Hence fn ∈ B
C (f ).

�

The following corollary is associated with the concept of ”accumulation
points” in classical topology.

Corollary 4.11. If (X, ξ) is a N-space, B is a subset of X, {xd : d ∈ D} is a
net in B. Then

(i) If {xd : d ∈ D} is N-convergent to x0 ∈ X, then x0 ∈ clξB.
(ii) In T − NEAR, for any x0 ∈ clξB, there is a {xd : d ∈ D} in B and

{xd : d ∈ D} is N-convergent to x0.

Proof. (i) If, to the contrary, x0 /∈ clξB. Then A = {{x0}, B} /∈ ξ. Since
{xd : d ∈ D}, and A

C = {X − {x0}, X − B}, there is an N ∈ D such
that for any n ≥ N , xn ∈ A

C(x0) = X − B. But this contradicts to
the assumption that {xd : d ∈ D} ⊆ B. Hence x0 ∈ clξB must be true.

(ii) In T − NEAR, from Theorem 4.8(i), N-convergence is equivalent to
topological convergence and clξB is the topological closure of the set
B. The conclusion must be true due to classical topology.

�

The following Corollary is a natural consequence of the Corollary 4.11:

Corollary 4.12. Let (X, ξ) be a N-space and B ⊆ X.

(i) If B is N-closed, then any convergent net in B must converge to a point
in B.

(ii) In T − NEAR, if the limit of any convergent net in B always remains
in B, then B is N-closed.



64 Z. Yang

The next several theorems show that the properties of being totally bounded,
compact or N2 are productive respectively.

Theorem 4.13. Let X be a set, (Y, η) be a N-space, and f : X → Y be a
N-preserving map. Then f −1(η) is totally bounded if and only if η is totally
bounded.

Proof. Assume that f −1(η) is totally bounded. We would like to show that η is
totally bounded by showing that it meets condition (B1). Arbitrarily take a B
such that ηB. Then f −1(B) /∈ f −1(η). There should be a finite subcollection
of B, say B0 ⊆ B, such that

⋂

f −1(B0) = φ. Hence
⋂

B0 = φ.
Now we assume that η is totally bounded. Arbitrarily take A /∈ f −1(η),

then ηf (A ). So there should be a finite subcollection of A , say A0 ⊆ A , such
that

⋂

f (A0) = φ. Now we can easily see that
⋂

A0 = φ. �

Theorem 4.14. Let {ξα : α ∈ Λ} be a family of nearness structures on X.
Let ξ = sup{ξα : α ∈ Λ}. Then ξ is totally bounded if and only if for each
α ∈ Λ, ξα is totally bounded.

Proof. First we assume that for each α ∈ Λ, ξα is totally bounded. Then for any
ξA , there should be finitely many ξαi , i = 1, 2, ...n as well as Aαi , i = 1, 2, ...n

such that Aαi /∈ ξαi and A ≺
n
∨

i=1
Aαi . Since each ξαi is totally bounded, each

Aαi contains a finite subcollection Bαi and
⋂

Bαi = φ.
n
∨

i=1
Bαi is a finite

subcollection of
n
∨

i=1
Aαi . We will claim that

⋂ n
∨

i=1
Bαi = ∅. Take an arbitrary

point x ∈ X, then for each i = 1, 2, ..., n, there is a Bxi ∈ Bαi such that x /∈ B
x
αi

.

So x /∈
n
∪

i=1
Bxαi . Hence x /∈

⋂ n
∨

i=1
Bαi . This shows that

⋂ n
∨

i=1
Bαi = φ. So ξ is

totally bounded.
Now we assume that ξ is totally bounded. Arbitrarily take a ξα. Then for

any ξαAα, we have ξAα. Since ξ is assumed to be totally bounded, Aα must
have finite subcollection with empty intersection. �

Theorem 4.15. If {ξα : α ∈ Λ} is a family of nearness structures on X . Its
product (

∏

α∈Λ Xα,
∏

α∈Λ ξα) is totally bounded if and only if each (Xα, ξα) is

totally bounded. Particularly, if (Y, η) is totally bounded and F ⊆ Y X where
X is a non-empty set, then (F , ξρ) is totally bounded.

Proof. The first conclusion can be deduced from Theorem 4.13 and Theo-
rem 4.14. The second conclusion can be deduced from the Theorem 3.18. �

The next Lemma, due to Herrlich ([8], 4.5 Proposition, Part (2)), will be
used in the proof of the following theorem.

Lemma 4.16. For a T-N space, the following conditions are equivalent:

(1) (X, ξ) is contigual;
(2) (X, ξ) is totally bounded;
(3) (X, ξ) is compact.



Pointwise convergence for nearness spaces 65

Theorem 4.17. If {ξα : α ∈ Λ} is a family of nearness structures on X. If
each (Xα, ξα) is compact, then its product (

∏

α∈Λ Xα,
∏

α∈Λ ξα) is also compact.

Proof. Recall that a compact nearness space satisfies condition (T) and (C).
It is easy to verify that the product is a T-N space, if each (Xα, ξα) is a T-N
space. According to Lemma 4.16, a T-N space is C-N if and only it it is totally
bounded. So the conclusion of this theorem follows from Theorem 4.15. �

Theorem 4.18. If {ξα : α ∈ Λ} is a family of nearness structures on X . If
each (Xα, ξα) is N2, then the product (X, ξ) = (

∏

α∈Λ Xα,
∏

α∈Λ ξα) is also N2.

Proof. By the definition of product of nearness structures, ξ = sup{P −1α (ξα) :
α ∈ Λ}, where Pα :

∏

α∈Λ Xα → Xα are natural projection maps. Let x, y ∈ X
and x 6= y, there should be at least one α ∈ Λ such that Pα(x) 6= Pα(y). Since
ξα is N2, there are Aα ⊆ X and Bα ⊆ X such that Aα ∩ Bα = ∅ and

Pα(x) ∈ Xα − clξα (Xα − Aα), Pα(y) ∈ Xα − clξα (Xα − Bα).

It is easy to see that P −1α (Aα) ∩ P
−1
α (αB) = ∅. We will try to show that

x ∈ X − clξ(X − P
−1
α (Aα)),

and
y ∈ X − clξ(X − P

−1
α (Bα)).

Take an arbitrary point

z ∈ P −1α (Xα − clξα (Xα − Aα)),

then
Pα(z) ∈ Xα − clξα (Xα − Aα).

and
{{Pα(z)}, Xα − Aα} /∈ ξα.

{{P −1α (Pα(z))}, P
−1
α (Xα − Aα)} /∈ P

−1
α (ξα).

{{z}, X − P −1α (Aα)} /∈ P
−1
α (ξα).

The last statement is true since

z ∈ P −1α (Pα(z))

and
P −1α (Xα − Aα) ⊆ X − P

−1
α (Aα).

Therefore,
z ∈ X − cl

P
−1

α (ξα)
(X − P −1α (Aα)).

This shows that

P −1α (Xα − clξα (Xα − Aα)) ⊆ X − clP −1α (ξα)(X − P
−1
α (Aα)).

Hence

x ∈ P −1α (Xα − clξα (Xα − Aα))

⊆ X − cl
P

−1

α (ξα)
(X − P −1α (Aα))

⊆ X − clξ(X − P
−1
α (Aα)).



66 Z. Yang

By similar argument,

y ∈ P −1α (Xα − clξα (Xα − Bα))

⊆ X − cl
P

−1

α (ξα)
(X − P −1α (Bα))

⊆ X − clξ(X − P
−1
α (Bα)).

Therefore, (X, ξ) is a N2- space. �

Now we will try to establish the relation between the compactness of the
underderlining set Y and a function space F ⊆ Y X .

Theorem 4.19. Let X be a set, and (Y, η) be a compact N-space. F ⊆ Y X .
Then

(i) The condition (a) is sufficient for (F , ξρ) to be compact.
(ii) If (Y, η) is also N2, then the condition (a) is also necessary for (F , ξρ)

to be compact.

(a) F is N-closed in (Y X , ξ∗).

Proof. (i) Since (Y, η) is a compact N-space. By Theorem 4.17, (Y X , ξ∗)
is compact. By Theorem 3.12, F ⊆ Y X , as an N-closed subset of a
compact space, is also compact under the subspace nearness structure.

(ii) If (Y, η) is a N2-space. By Theorem 4.18, (Y
X , ξ∗) is an N2- space.

Then by Theorem 3.17, (F , ξρ), as a compact subspace of an N2- space,
is N-closed.

�

Theorem 4.20. If X is a set, and (Y, η) is an N-space. F ⊆ Y X . Then

(i) the conditions (a) and (b) are sufficient for (F , ξρ) to be compact.
(ii) If (Y, η) is also N2, then the conditions (a) and (b) are also necessary

for (F , ξρ) to be compact.

(a) F is N-closed in (Y X , ξ∗)
(b) For any x ∈ X, F [x] = {f (x) : f ∈ F } is contained in a compact

subspace of (Y, η).

Proof. (i) Assume that (Yx, ηx) is a compact nearness subspace of (Y, η)
with ηx = η|Yx and F [x] ⊆ Yx ⊆ Y . Then F ⊆ (

∏

x∈X
Yx,

∏

x∈X
ηx)

and the later space is compact, according to Theorem 4.17. It is easy
to see that ξρ =

∏

x∈X ηx|F . Since F is N-closed also, it is N-compact
due to Theorem 3.12.

(ii) If (Y, η)is a N2-space, and (F , ξρ) is compact. It follows from Theo-
rem 3.17 that (a) is true. And since the evaluation map ex : (F , ξρ) →
(Y, η) is N-preserving. By Lemma 3.14, F [x] = {f (x) : f ∈ F } =
{ex(f ) : f ∈ F } is also compact.

�



Pointwise convergence for nearness spaces 67

Summary. This paper essentially lays the foundation for some possible ap-
plications of the theory of nearness function spaces in digital topology. The
main results is the introduction of the pointwise convergent nearness spaces
in the function spaces in such a way that is consistent with the existing and
established structures. Two Ascoli’s type of theorems on nearness spaces are
established also.

Any deformation of a digital image (such as thinning) can be considered as
a function from the digital plane to itself. Of course, we would prefer those
functions to preserve some properties of the digital image, such as ”nearness”.
When a sequence of deformations are applied to a digital image, we would like
to be able to make some type of projection or prediction about the final images
based on the type of deformations involved in the sequence. We believe that the
line of work presented in this paper will be helpful to address those issues.

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[20] J. Adàmek, H. Herrlich and G. E. Strecker, Abstract and Concrete Categories, The Joy
of Cats, (John Wiley and Sons, New Work 1970).

Received May 2008

Accepted October 2008

Zhanbo Yang (yang@uiwtx.edu)
Department of Mathematical Sciences, University of the Incarnate Word, 4301
Broadway, San Antonio, TX 78209, USA