1. Introduction
The theory of bounded sets on metric spaces has been studied by many authors with different motivations. For instance, Kubrusly and Willard proved that a metric space
is totally bounded if and only if every sequence in X has a Cauchy subsequence. In 2012, Olela Otafudu investigated total boundedness of the u-injective hull of a totally bounded T0-ultra-quasi-metric space. He first defined a set to be bounded if it is contained in a double ball and total bounded if it is contained in the union of finite number of
-open balls. He then proved that total boundedness is preserved by the ultra-quasimetrically injective hull of a T0-ultra-quasi-metric space (see ( [1], Proposition 5.4.1)).
According to Cobzas ( [2], p. 63), a quasi-pseudometric space
is said to be totally bounded if for each
there exists a finite subset
of X such that
. As it is known, in metric spaces precompactness and total boundedness are equivalent notions, a result that is not true in quasi-metric spaces (see ( [2], Proposition 1.2.21)). In quasi metric spaces, Mukonda and Otafudu have defined a set to be Bourbaki bounded if for each
and a nutural number n, there exists a finite subset
of X such that
.
Morever, our recent work [3] has extended the concept of bornology from metric settings to the framework of quasi-metrics. Naturally, this has led to the speculation of what is the relationship between the bornology of bounded sets and other types of bornologies on quasi-metric spaces. Toachieve this, a careful study of bornologyof bounded sets, bornology of totally bounded sets and bornolgies of bourbaki bounded sets in quasi-pseudometric spaces is required.
In this present work, we intend to generalize some classical bornological results of Garrido and Meroño [4] on classes of bounded sets from metric spaces to the category of quasi-metric spaces. For instance, given a compatible quasi-metric, we intend to give some necessary and sufficient conditions for which a bornology of totally bounded sets and bornology of bourbaki bounded sets coincide with our quasi-metric bornology studied in [5].
2. Preliminaries
This section recalls and introduces the terminology and notation for quasi-metric spaces we will use in the sequel. Further details about theory of asymmetric topology can be found in [2] [6] [7].
Definition 2.1. Let X be a set and let
be a function mapping into the set
of the nonnegative reals. Then, q is called a quasi-pseudometric on X if
1)
whenever
.
2)
whenever
.
We say q is a T0-quasi-metric provided that q also satisfies the following condition:
If q is a quasi-pseudometric on a set X, then
defined by
for every
, often called the conjugate quasi-pseudometric, is also quasi-pseudometric on X. The quasi-pseudometric on a set X such that
is a pseudometric. Note that if
is a quasi-metric space, then
is also a metric.
Remark 2.2. [2] Let
be a quasi-pseudometric space. The open ball of radius
centred at
is the set
. The collection of open balls yields a base for the topology
and it is called the topology induced by q on X. Similarly, the closed ball of radius
centred at
is the set
. If
is a quasi-pseudometric space, then the pair
where
and
is called a double ball. In general,
, with
and
, is called the family of double balls.
Note that the set
is a
-closed set, but not
-closed in general. The following inclusions holds:
Definition 2.3. ( [3], Definition 4.1) Let
be a quasi-pseudometric. An arbitrary subset A is called q-bounded if only if there exists
,
and
such that
.
Definition 2.4. Let
be a quasi-pseudometric space and
. We say that F is totally bounded, if for any
there exists a finite subset
of X such that
Definition 2.5. Let
be a quasi-pseudometric space and
. We say that F is q-Bourbaki-bounded, if for any
there exists a finite subset
of X and for some positive integer n such that
Definition 2.6. A bornology on a set X is a collection
of subsets of X which satisfies the following conditions:
1)
forms a cover of X, i.e.
;
2) for any
, and
, then
;
3)
is stable under finite unions, i.e. if
, then
If we take a nonempty set X and a bornology
on X, then the pair
is called a bornological universe. For every nonempty set X, the family
is the smallest bornology on X.
Recall from [3] that the bornology of quasi-pseudometric bounded sets is denoted by
. However, in [8], the family of totally bounded subsets and boubark bounded sets their bornologies are denoted by
and
respectively. We will compare these bornologies in the next sections.
Let
be a T0-quasi-metric space. Then
is called bicomplete provided that the metric space
is complete. A mapping f between two quasi-metric spaces
and
is said to be quasi-isometry if
for all
in X.
A bicompletion of a quasi-metric space
is a bicomplete quasi-metric space
in which
can be quasi-isometrically embedded as a
-dense subspace.
We recall the concepts of asymmetric norms and semi-Lipschitz functions in quasi-metric spaces.
Definition 2.7. [2] An asymmetric norm on a real vector space X is a function
satisfying the conditions:
1)
then
;
2)
;
3)
,
for all
and
. Then the pair
is called an asymmetric normed space.
The conjugate asymmetric norm
of
and the symmetrized norm
of
are defined respectively by
An asymmetric norm
on X induces a quasi-metric
on X defined by
If
is a normed lattice space, then the function
with
is an asymmetric norm on X.
Definition 2.8. Let
be a quasi-metric space and
be an asymmetric normed space. Then a function
is called k-semi-Lipschitz (or semi-Lipschitz) if there exists
such that
(1)
A number k satisfying inquality (1) is called semi-Lipschitz constant for
.
3. Some Results of Boundedness in Quasi-Metric Spaces
This section is as a result of the distinction that we gave in [3] about the bornologies
and
. We will investigate further the connection between the bornologies
,
,
and
.
Lemma 3.1. If
is a quasi-metric space. Then the following statement is true:
(2)
and the quasi-metric bornologies
and
are equivalent.
Proof. Let
, then A is
-bounded. By Remark 2.2, A is q-bounded too. Thus
. The equivalence of
and
comes from the fact that any subset A of X is q-bounded if and only if it is
-bounded. □
The converse of Lemma 3.1 above does not holds. i.e., a set on a quasi-metric can be q-bounded but not
-bounded (check ( [3], Remark 4.2)).
Definition 3.2. ( [6], p.85) Let
be a T0-quasi-metric space. Then
is called joincompact provided that the metric space
is compact.
Theorem 3.3. (Compare ( [9], Theorem 3.78).) Let
be a T0-quasi-metric space. A set
is joincompact if and only if B is both bicomplete and totally bounded.
Proof. We leave this proof to the reader. □
We rephrase the above theorem in the following Corrolary as proved by Fletcher and Lindgreen in quasi-uniform spaces (see ( [7], p. 65)).
Corollary 3.4. ( [7], Proposition 3.36) Let
be a T0-quasi-metric space. Then
is totally bounded if and only
is compact.
Definition 3.5. ( [2], Definition 1.44) Let
be a T0-quasi-metric space. Then
is called supseparable provided that the metric space
is separable.
Proposition 3.6. (Compare ( [9], Proposition 3.72)) A totally bounded quasi-pseudometric space
is supseparable.
Proof. Suppose
is totally bounded, for any positive interge n, we can
find a finite set
such that for all
,
. Now let
. The set B is either finite or infinitely countable, thus countable. To show the
-density of B, let us pick
, then we have
implying that
and
. This
proves that x is a
-limit point of B and hence B is a
-dense subset of X. Consequently,
separable and by Definition 3.5,
is supseparable. □
The next example shows that for finite dimension spaces, total boundedness coincide with boundedness.
Example 3.7. If we equip a real unit interval
with the T0-quasi-metric
, then the pair
is both q-bounded and totally bounded space.
Proof. It can be seen that X is q-bounded. Now If we pick
to be a finite subset of
and
, then
□
The next Lemma proves that for infinite dimension spaces, total boundedness and quasi-metric boundedness are two different notions.
Lemma 3.8. Let
be a quasi-metric space, then
.
Proof. Let
. For
, there exists a finite subset
of B such that
. The set B is a finite family of
-bounded subsets thus its is
-bounded. Hence
by Lemma 3.1. □
The following example illustrates the converse of Lemma 3.8 above.
Example 3.9. Let us equip the set of natural numbers
with the T0-quasi-metric
The T0-quasi-metric space
is q-bounded but not q-totally bounded.
Proof. For all
we can find
such that
. But any finite set
with the discrete metric
, the set
can not be covered by
for
. Hence,
is not q-totally bounded. □
It is important to note that
is a metric bornology in the sense of Beer et al. [10].
Definition 3.10. Let
be a quasi-pseudometric space and
. For any
, we define the
-enlargement
of F by
and
Furthermore,
Remark 3.11. For a given quasi-pseudometric space
. For any
and
. It is easy to see that if
is a
-chain in
of length n from x to y, then
is also a
-chain in
and in
of length n from x to y. We have
(3)
and
(4)
Lemma 3.12. Let
be a quasi-pseudometric space and for any
. We have
.
Lemma 3.13. Let
be a quasi-pseudometric space and
. For any
and
, we have
Corollary 3.14. Let
be a quasi-pseudometric space and
. If there exists a
-chain of length n from x to y in
, then there exists a
-chain of length n from y to x in
whenever
.
Lemma 3.15. If
is a quasi-metric space. Then the following statement is true:
(5)
and the quasi-metric bornologies
and
are equivalent.
Proof. Let
. Suppose that
. Then there exists a finite set
such that
for some positive integer n. By inclusion (3) we have
for some positive integer n. Hence
. Note that Corollary 3.14 confirms the equivalence of
and
. □
The converse of the above lemma does not always hold. Let us determine this from the following example.
Example 3.16. Consider the four point set
. If we equip X with T0-quasi-metric q defined by the distance matrix
that is,
whenever
. The symmetrized metric
of q is induced by the matrix
Let
. If we consider the sequence
. Then we have
Hence the sequence
is a
-chain in
of length 2 from 4 to 1. But the same sequence
is not a
-chain in
of length 2 from 4 to 1 because
We state the following lemma that we will use in our next proposition.
Lemma 3.17. Let
be a quasi-pseudometric space. For some positive integer n,
and
, we have
Proof. Let
, then for some j with
,
. Moreover, for some j with
, there exists
a
-chain of length n from
to y such that
,
and
for all i with
. Furthermore, we have
Thus, for some j with
,
. Hence,
. □
Proposition 3.18. Given a quasi-pseudometric space
. If F is a subset of X and
, then we have the following conditions:
1)
.
2)
.
Proof.
1) Let
. Suppose
then there exists a set
such that
for some positive integer
. Therefore,
.
2) Since
there exists a set
and some positive integer n such that for
we have
. By Lemma 3.17,
. Hence,
. □
Let us provide the summary of the connections between these bornologies in the following remark.
Remark 3.19. If
is a quasi-pseudometric space, then we have the following inlusions:
But if
is an asymmetric normed space, then we have
We have provided the proof in Proposition 4.1.
4. Main Results on Bornologies
One would still wonder, if is it indeed posible to find a quasi-metric metric
equivalent to q such that
or
.
Proposition 4.1. Suppose that
is an asymmetric normed space. Then we have the following:
Proof. For
follows from Proposition 3.18 (b).
For
, suppose that F is
-bounded then
for some
and
. For any
, there exists
such that
.
Let
. We define
whenever i with
and
. Then
Thus, for any
we have obtained a
-chain of length n on
from
to f. Therefore,
. □
Definition 4.2. [11] Given a Hilbert cube
, the product topology is defined in a usual way by a quasi-pseudometric
where
.
Theorem 4.3. ( [11], Theorem 3.10) Every supseparable quasi-metric space is embeddable as subspace of the Hilbert cube
.
Theorem 4.4 (Tychonoff’s Theorem). The topological product of a family of compact spaces is compact.
Theorem 4.5. (Compare ( [10], Theorem 3.1).) Let
be a quasi-metric space and let
. The following conditions are equivalent:
1) There exists an equivalent quasi-metric
such that
.
2) The quasi-metric space
is supseparable.
3) There is an embedding
of X into some quasi-metrizable space Y such that the family
is cofinal in
.
4) There exists an equivalent quasi-metric
with
.
Proof.
1
2: If there exists an equivalent quasi-metric space
such that
, then
where
are
-totally bounded subsets. This means that X is a countable union of
-totally bounded sets, thus its
-totally bounded and by Proposition 3.6, the quasi-metric space
is supseparable.
2
3: First case: If q is bounded, then by Theorem 4.3, we can find an embed
ding
. Let
and choose
so that
. Since
is joincompact with
respect to product topology, its subset Y is joincompact and confinal in
.
Second case: If q is unbounded, consider
as a
-dense subset in X. For each i in
, Let us define
by
. Now if A is a nonempty
-closed subset of X and
then we can choose
with
and
. From the choice of
, the set
separates points from
-closed sets and we can define an embedding
by
equipped with the product topology.
Now let p be a quasi-metric compatible with the product topology on
, we
now prove that
equipped with the relative topology is
cofinal in
. If
is chosen arbitrary then for each
,
is q-bounded, so by the Theorem 4.4,
is joincompact as it is contained in a product
. Suppose
is not confinal
in
. Let
then for each
, take
and pick
with
and
.
By the joincompactness of B and the quasi-metrizability of Y, we can find some
-subsequence
of
such that
. This implies that
. But this is not possible, since q is unbounded.
3
4: If
is an quasi-metric equivalent to q then
by ( [3], Theorem 5.4). To prove that
, let
and
be a bicompletion of
. Since
is bicomplete by, the set
is compact. Given the cofinality of
, let us choose
with
. But this means that
Thus
and it follows that
. For the reverse inclusion. If
, we can choose
with
.
The
is compact and
-totally bounded.
Therefore,
. The equivalence 4
1 follows from ( [3], Theorem 5.4). □
Definition 4.6. (Compare ( [10], Definition 3)). Let
be a T0-quasi-metric space. Given the point
and a quasi-metric bornology
on X we can form the one-point extension of X associated with
by a
.
If
is the topology X, then the corresponding topology on
is defined by
The quasi-metric bornology associated with
is denoted by
.
Remark 4.7. If
is a
-closed base of the bornology then
forms a
-neighbourhood base at the point p.
Lemma 4.8. Let
be a T0-quasi-metric space. If the bornology
is quasi-metrizable then the associated bornolgy
on
is quasi-metrizable.
Theorem 4.9. (Compare ( [10], Theorem 3.4)) Let
be a quasi-metric space The following conditions are equivalent:
1)
has a countable base;
2) There exists an equivalent quasi-metric
such that
3) The one-point extension of X associated with
is quasi-metrizable.
4) The one-point extension of X associated with
has a
-neighborhood base at the ideal point.
Proof.
1
2: Since
has a countable base by Hu’s theorem (see ( [3], Theorem 4.18)) there exists an equivalent quasi-metric
such that
.
2
3: By (2),
. From Lemma 4.8
on
is quasi-metrizable thus
is quasi-metrizable.
3
4 Since the bornology
has a
-closed base, thus by the Remark 4.7
has a
-neighborhood base at the ideal point.
4
1: If
have a
-neighborhood base at each point, then
has countable base. □
Definition 4.10. Let
be a quasi-metric space and
be an asymmetric normed space. A function
is called semi-Lipschitz in the small if there exists
and
such that if
then
.
The following lemma follows directly from the definitions of semi-Lipschitz in the small function and uniformly continuous.
Lemma 4.11. Let
be a quasi-metric space and
be an asymmetric normed space. If a function
is semi-Lipschitz in the small, then
is uniformly continuous.
Theorem 4.12. (Compare ( [12], Theorem 3.4)) Let
be a quasi-metric space and
. Then the following conditions are equivalent:
1)
;
2) if
is an asymmetric normed space and
is uniformly continuous, then
;
3) if
is an asymmetric normed space and
is semi-Lipschitz in the small function, then
;
4) if
is semi-Lipschitz in the small function, then
.
Proof.
(1)
(2) If
is uniformly continuous then there exists
such that whenever
with
, we have
(6)
By the q-Bourbaki-boundedness of F, there exists
such that
for some positive integer n. If we take f artbitrary in F, then there exists k with
such that
. Then for some k with
, there exists a
-chain
with
,
and
(7)
It follows from the uniform continuity of
and inequality (6) that
(8)
Hence, for some k with
, we have
Thus,
for any
and
. Therefore,
is
-bounded.
(2)
(3) Follows from Lemma 4.11.
(3)
(4) Follows directly by replacing
with
in (3).
(4)
(1). Suppose that F is not q-Bourbaki-bounded. Then there exists a
such that if
and a positive integer n, we have
. We have two cases on the structure of F.
Case 1: If
, then there exists a positive integer n such that for all
Let
be an arbitrary point of F. We choose a positive integer
such that
Since F is not q-Bourbaki-bounded, there exists
such that
. It follows that
by the choice of
.
One chooses another
such that
and
. Moreover, since
, we can find
. Continuing this procedure by induction, we can find a sequence (
) with distinct terms in F such that for any
we have
. Therefore, we define a function
by
It follows that the function
is constant on
and it is unbounded on F since
. Therefore, the function
is semi-Lipschitz in the small function.
Case 2: If there exists
and for all positive integer n, there exists
such that
For
, let
be the smallest positive integer n such that
(9)
We then define the function
by
By definition, the function
is unbounded on F. We now have to show that if
and
, then for
If either x or y is not related to f with respect to
, then since
, both x and y are not related to f with respect to
and
If
and
, then we have some cases on
and
:
If
. Suppose that
then
and
which implies that
hence
.
Furthermore,
If
and
, then
If
and
(i.e.,
) with
, then there is nothing to prove since
.
If
, then
Furthermore,
Since
is the smallest n such that
, it therefore means
Thus, we have
(10)
We claim that,
(11)
Suppose otherwise, i.e.,
, then
So
which implies that
but this is a contradiction since
.
Combining (10) and (11) we have
Therefore, the proof is complete. □