1. Introduction
An important class of spaces was introduced by I.A. Bakhtin (under the name almost metric spaces) and rediscovered by S. Czerwik (under the name "b-metric spaces"). b-metric spaces generalize “classic” metric spaces by replacing in the definition of a metric the triangularity axiom for all with a more general axiom for all where is a fixed constant. The class of b-metric spaces includes such interesting and important for applications cases, as , where defined by or defined by . A series of b-metrics can be obtained from an ordinary metric by the following construction. Let be a fixed constant and let be a continuous increasing mapping, such that and for all . Further, let be a metric on a set X, Then by setting we get a b-metric on the set X, which, in case fails to be a metric.
Recently there were published several papers where the induced topology of b-metric spaces was applied. Unfortunately, in most papers the authors assumed that the open balls
are "really open", which is open in the induced topology. However, as it was noticed in [
1] and soon rediscovered also in [
2,
3], it is generally not true. Some recent results concerning b-metric spaces can be found e.g., in [
4,
5,
6,
7].
To “remedy” this “shortage” of b-metric spaces, in [
1], Kirk and Shahzad introduced the class of strong b-metric spaces (in this paper we call them sb-metric spaces). In the definition of an sb-metric space the third (triangularity) axiom is given by the inequality
for some
. Obviously, the class of sb-metric spaces lies in between the class of metric and the class of b-metric spaces. As shown by Kirk and Shahzad [
1], in sb-metric spaces open balls are really open in the induced topology. Thanks to this fact, sb-metric spaces have many useful properties common with ordinary metric spaces. Unfortunately, Kirk and Shahzad in [
1] could not present any examples of strong b-metric spaces that fail to be metric. It was the first, original, aim of this paper to present a series of such examples, and this is done in
Section 5.
However, having this primary aim in mind, we decide that it could be interesting, and probably useful, to develop our work on the basis of the operation ⊕, which is actually a kind of extended t-conorm. This allows for considering sb-metric spaces, and parallelly ordinary metric spaces and b-metric spaces in two ways: as ordinary metrics, if ⊕ in the triangularity axiom is an addition (that corresponds to the extension of Łukasiewicz t-conorm considered on the triangle ) and as ultra-metrics, in case when ⊕ is maximum (that corresponds to the maximum t-conorm).
The structure of the paper is a follows. In
Section 2, we define the notion of an extended t-conorm. Some properties of extended t-conorms needed in the research are highlighted and examples are given. The
Section 3 contains definitions of different metric-type operations and the related metric-type spaces. These spaces are reconsidered in the
Section 4 from the categorical point of view.
Section 5 contains a list of examples of non metric sb-metric spaces with different properties as well as examples of some metric-type spaces on the basis of extended t-conorms.
Section 6 is devoted to the study of products and coproducts of metric-type spaces, specifically to products and coproducts of sb-metric spaces. In
Section 7, we discuss some prospect for future work.
2. Extended T-Conorms
Definition 1. Let . A binary operation will be called extended t-conorm if, for all , the following properties hold:
(⊕1) ⊕ is commutative, that is
(⊕2)⊕ is associative, that is
(⊕3) ⊕ is monotone, that is
(⊕4) 0 is the neutral element for ⊕, that is
Remark 1. Note that in case operation ⊕
is defined on and it takes its values in , then the definition of extended t-conorm reduces to the concept of a t-conorm [8]. Just for this observation, we refer to ⊕
as an extended t-conorm. We will also need the following special properties of operation ⊕:
Definition 2. ⊕ is called distributive, if for all
(⊕5) .
⊕ is called compressible, if,
(⊕6)
⊕ is called continuous at the bottom level, if this operation is continuous in all points of
(and, hence, by symmetry also on .
Below, we give some examples of extended t-conorms ⊕, which are mentioned in the paper.
Example 1. Let Thus, is an ordinary addition. It is obvious that + satisfies all properties from Definition 1. One can easily see that this operation is distributive, compressible, and is continuous on the whole space . When restricted to the triangle , operation reduces to the Łukasiewicz t-conorm.
Example 2. Let , where ∨ denotes the maximum. It is obvious that satisfies all properties from Definition 1. Thus, ⊕ is the extension of the maximum t-conorm from to . One can easily see that this operation is distributive and it is continuous on the whole space . The compressibility of ∨ follows from the next easily provable lemma:
Lemma 1. if and only if
Example 3. Let It is obvious that satisfies properties , and from Definition 1.
We verify the associativity of , as follows. The continuity of on the whole is obvious. Unfortunately, is not distributive:and does not satisfy the property . For example but . Example 4. We define operation by setting . Obviously it is just the product t-conorm on the closed interval, and hence the properties of Definition 1 are valid.
We do not know whether it is possible to extend the product t-conorm to the whole square preserving properties .
Example 5. The h-shifted arithmetic sum, where is a fixed constant and it is defined asand satisfies all properties from Definition 1. However, it is not distributive, sinceand is not continuous at any point on . 3. Metric-Type Structures Based on Extended T-Conorms
To make our exposition coherent, in the following definition, we specify the terminology concerning metric and its generalizations that can be found in the literature (see e.g., [
9,
10,
11]):
Definition 3. A mapping is called semi pseudometric if it satisfies the following two properties
(m1)
(m2)
A semi-metric is a semi pseudometric satisfying the following strengthen version of the first axiom:
(m1’)
Definition 4. Given an extended t-conorm , a semi (pseudo)metric is called
(m3) ⊕-based (pseudo)metric, or just ⊕-(pseudo)metric, if
(mb3) ⊕-based b-(pseudo)metric, or just ⊕-b-(pseudo)metric if there exists such that
(msb3) ⊕-based sb-(pseudo)metric or just ⊕-sb-(pseudo)metric, if there exists , such that
.
In particular, in case constant k is fixed in conditions and , we may speak, respectively, of ⊕-bk-(pseudo)metric and ⊕-sbk-(pseudo)metric.
Remark 2. In case , that is an arithmetic addition, ⊕
-(pseudo)metric, ⊕
-b-(pseudo)metric and ⊕
-sb-(pseudo)metric reduce to the definitions of an ordinary (pseudo)-metric, b-(pseudo)metric [12,13,14] and sb-(pseudo)-metric [1] (Actually Kirk and Shahazad [1] call this kind of a mapping by strong b-metric. However, we consider this term to be inappropriate, since it may lead to a misunderstanding when extending this notion to the case of fuzzy metric-type structures since it comes into collision with the concepts of a strong fuzzy metric and a strong b-fuzzy metric widely used in the literature). In this case, we usually omit sign ⊕.
In case , which is maximum, ⊕
-(pseudo)metric reduces to the definition of an ultra-(pseudo)-metric, see, e.g., [15]. Accordingly, ⊕
b-(pseudo)metric and ⊕
-(pseudo)metric in this case could be called ultra-b-(pseudo)metric and ultra-sb-(pseudo)metric, respectively. By applying other extended t-conorm based operators, we obtain new versions of metric-type structures some of which will be considered in this paper.
Having these general definitions for different types of metric-type structures, in the sequel we will mainly be interested in ⊕- sb-(pseudo)metrics, since just sb-(pseudo)metrics are the main purpose of this paper. However, for comparison, we will sometimes comment the versions of the obtained results for ⊕-sb-(pseudo)metrics in case of the ⊕-(pseudo)metrics from one side and in the case of b-(pseudo)metrics from the other. In particular, we will be interested in what properties of an extended t-conorm ⊕ are important for the validity of different properties of metric-type structures.
In order to make exposition more homogeneous, we will restrict to the case of ⊕-sb-metrics, which is with assumption of axiom . A reader can easily reformulate and verify the validity of our results for the case of ⊕- sb-pseudometrics, which is without the assumption of axiom .
4. Categories of ⊕-Metric-Type Spaces
To view the ⊕-metric-type spaces considered in the previous section in the framework of a category, we have to define their morphisms, which is "continuous’ mappings. To do this, we first define an open ball in a ⊕-metric type space.
Definition 5. Given a set X, an extended t-conorm ⊕
and an ⊕
-based metric-type structure . We define an open ball with center and radius in the space as follows:We call an open ball really open, if, for every , there exists , such that . Note that saying that a ball is really open just means that it is open in the topology obtained from the family as a base.
Theorem 1. Let ⊕ be a continuous at the bottom extended t-conorm. Subsequently, open balls in ⊕-metric spaces and in ⊕-sb-metric spaces are really open.
We prove the theorem in the case of ⊕-sb-metric spaces. Patterned after this proof, a reader can easily prove it for ⊕-metric spaces. Certainly, this fact for ⊕-metric spaces in cases of ordinary metrics () and ultrametrics () is well known.
Proof. Consider the open ball
where
and
. Let
. Then
and there exists
, such that
since ⊕ is continuous on
. (Indeed, consider
. Notice that
and by assumption
is continuous at 0. Therefore, there exists
, such that
). Now, we shall show that
. Let
. Subsequently, we have
and this means that
. □
Remark 3. The statement of the previous theorem actually means that open balls are open in the topology generated by the base . This might not be true for ⊕
-b-metric spaces, even in case , see e.g., [2,3]. Corollary 1. In case ⊕ is continuous at the bottom, ⊕-metric and ⊕-sb-metric spaces are first countable.
Theorem 2. Let ⊕ be a continuous at the bottom extended t-conorm. Subsequently, closed balls defined as in ⊕-metric spaces and in ⊕-sb-metric spaces are closed.
Proof. We prove the theorem in the case of ⊕-sb-metric spaces. Let
d be a ⊕-
metric on
X,
where
,
. We shall show that
, where
. For any
, we have
and
means that
. Therefore,
is open and
is closed in
X. □
Definition 6. Two ⊕-sb-metrics (⊕-b-metrics) are called equivalent if they induce the same topology.
Remark 4. Because the topologies induced by ⊕-sb-metrics are first countable, topologies and coincide if and only if a sequence converges to a point in topology if and only if converges to in topology
Definition 7. Let ⊕ be a continuous at the bottom extended t-conorm and let and be ⊕-metric or ⊕-sb-metric spaces. A mapping is called continuous if it is continuous as a mapping where to and are the topologies induced by and respectively.
Because, in these cases, open balls are really open and hence form bases for the corresponding first countable topologies, we easily get the following result:
Theorem 3. The following are equivalent for a mapping where and are ⊕-metrics or ⊕-sb-metrics and ⊕ is a continuous extended t-conorm:
- 1.
is continuous;
- 2.
for every , every there exists such that whenever
- 3.
if a sequence converges to a point in the space , then the sequence converges to the point in the space .
Theorem 4. If the extended t-conorm ⊕ is continuous, then an ⊕-sb-metric of the space is continuous as the mapping .
Proof. Because the topology induced by an ⊕-sb-metric is first countable, it is sufficient to prove that, if
are sequences in
X and
,
, then
. We do it as follows:
Taking limits on the both sides of the inequality, we conclude first the existence of
and then that
□
Now we are ready to define some categories for ⊕-metric type spaces under the assumption that ⊕ is a continuous extended t-conorm.
The objects of the category ⊕-Metr are pairs where X is a set and d is an ⊕-metric on it. The morphisms of the category ⊕-Metr are continuous mappings .
The objects of the category ⊕-Mtrz are pairs where X is a set and is a topology induced by some ⊕-metric d. The morphisms of the category ⊕-Mtrz are continuous mappings .
The objects of the category ⊕-SbMetr of ⊕-sb-metric spaces are pairs where X is a set and d is an ⊕-sb-metric on it. The morphisms of the category ⊕-SbMetr are continuous mappings .
By ⊕-SbkMetr, we denote the full subcategory of the category ⊕-SbMetr, whose objects are ⊕-sbk-metric spaces.
The objects of the category ⊕-SbMtrz are pairs , where X is a set and is a topology that is induced by some ⊕-sb-metric d. The morphisms of the category ⊕-SbMtrz are continuous mappings . Let ⊕-SbkMtrz be the full subcategory of ⊕-SbMtrz whose objects are obtained by ⊕-sbk-metrics.
On the basis of ⊕-b-metrics we introduce two categories.
- (a)
(see [
3]) Let
be the family of all unions of open balls, that is
The family
is obviously a supratopology (see e.g., [
16,
17]), that is closed under taking arbitrary unions.
need not be a topology: the intersection of even two elements
need not be in
since an open ball need not be open in
. Let ⊕
-b-MetrS be a category whose objects are pairs
where
X is a set and
d is an ⊕-b-metric on it and whose morphisms are continuous mappings
- (b)
(see [
3]) We call a set
-open if for every
there exists
, such that
One can easily notice that
and the union of any family of
-open sets is
-open. Thus
is indeed a topology on
X.
Obviously, each can be expressed as a union of some open balls and for this reason it belongs to . Thus, On the other hand not every open ball needs to be -open and, hence, generally Let ⊕-b-MetrT be the category whose objects are pairs where X is a set and d is an ⊕-b-metric on it and whose morphisms are continuous mappings .
5. Examples
5.1. Examples of Sb-Metric Spaces
In this subsection, we assume that
. Henceforth the resulting spaces are “classical” sb-metric spaces [
1].
Example 6. Let where and be a distance function defined as follows:Then d is an sb4-metric. Proof. Notice that .
For any
such that none of the pairs
is consecutive it holds
Accordingly, we only need to investigate those and z for which at least one of the pairs is consecutive.
Let be given. Without the loss of generality, we assume that :
If or , then the equality is obvious.
If
, then it means that
x and
z are not consecutive and we have
and
If
, then it means that
z and
y are not consecutive and we have
and
If , then it means that x and y are not consecutive and . Here, we have three cases.
Case 1: x and z are consecutive, but y and z are not consecutive.
For this case,
since
x and
z are consecutive. Let
. Here, notice that
. Subsequently,
Case 2: x and z are not consecutive, but y and z are consecutive.
Let
and
. Here, notice that
. Afterwards,
Case 3: x and z are consecutive and y and z are consecutive.
For this case again
since
x and
z are consecutive. Let
and
. Subsequently,
□
Remark 5. In above example, d is not a metric. To see this, consider and . Afterwards,On the other hand, the induced topology is the same as the topology induced on X by the ordinary topology on the real line. More precisely , where is the ordinary metric. Hence, is not a discrete space. However, any element X except 0 is an isolated point. Example 7. Let and . We denote where and . Define , as follows:Subsequently, d is an sb3-metric. Proof. For any
, we will show that
We need to consider five cases:
Case 1: if , then it is obvious since is usual metric on each i-level.
Case 2: if
and
, where
, then
and
Case 3: if
and
where
, then
and
Case 4: if
and
where
, then
and
Case 5: if
and
, where
, then we patterned the solution of Example 2.1 in [
18]. Let fix
and
(note that other combinations would give the same results). Subsequently, we need to check the following:
□
Remark 6. In the above example, notice that every i-level is an ordinary metric and, between the levels, it behaves as sb3-metric. Moreover, the induced topology does not contain any isolated point.
Example 8. Let X be the unit disc and be the unit circle in with the center in the origin of the plane and let the distance function be defined, as follows: for and where is the post office metric, which is . Then d is an sb2-metric which fails to be a metric. Proof. Because other cases are obvious, the only case that we need investigate is the case when
and
.
□
Remark 7. This example can be generalized for any other value of if we define for .
5.2. Examples of ⊕-Metric Type Spaces
Example 9. Consider the discrete space where the distance function d defined, as follows:
Notice that d is neither a metric nor an ultra metric.
- (1)
If ,then d is an ⊕-sb2-metric (simply sb2-metric).
- (2)
If then d is an ⊕-sb4-metric (might be called ultra sb4-metric).
- (3)
If (see Example 2.7) then d is an ⊕-metric. Note that is not distributive. However, in this case, distributivity is not a necessary condition.
- (4)
Let ⊕ be the h-shifted arithmetic sum for an arbitrary h. If , then d is an ⊕-metric. If , then d is an ⊕-sb- -metric.
Example 10. Let and . Define as follows: - (a)
If , then d is an ⊕-metric.
- (b)
If , then d is an ⊕--metric.
- (c)
If , then d is an ⊕--metric.
Proof. The case (a) is obvious.
(b) Let
. Then the only cases we need to investigate are whether the equalities
and
hold. We get them, as follows:
and
Therefore,
d is an ⊕-
-metric space. On the other hand,
and
Hence, d is neither an ⊕-metric and nor an ⊕--metric.
(c) Let
. Subsequently, the only case we need to investigate is
Hence,
d is neither an ⊕-metric and nor an ⊕-
-metric. However,
and, hence,
d is an ⊕-
-metric. □
Example 11. Now, we consider the above example where ⊕ is h-shifted arithmetic sum.
- (a)
If , then d is an ⊕-metric.
- (b)
If and , then d is an ⊕-metric.
- (c)
If and , then d is an ⊕--metric.
- (d)
If and , then d is an ⊕-metric.
- (e)
If and , then d is an ⊕- metric.
Proof. The case (a) is obvious, since .
(b) Let
and
, then we have
and
Therefore,
d is an ⊕-metric.
(c) Let
and
, then we have
and
Hence, d is an ⊕--metric.
(d) Let
and
, then we have
means that
d is an ⊕-metric.
(e) Let
and
, then we have
Consequently,
d is an ⊕-
-metric. □
Remark 8. Similarly, it can be shown that d is not an ⊕--metric and d is not an ⊕--metric in the cases (c) and (e) in the above example, respectively.
Here, we want to emphasize that continuity of ⊕ at the bottom level is essential for the openness of the open balls for ⊕-sb metric spaces as well as for ⊕-metric spaces.
Example 12. Let . Define , as follows: If we choose ⊕ as the h-shifted arithmetic sum for , then by Example 11, d is an ⊕-metric. However, the open balls where are not open in the induced topology.
6. Products of ⊕-Sb-Metric Spaces
Here, we distinguish the cases of finite and infinite (countable) number of spaces.
6.1. Products of Finite Families of ⊕-Sb-Metric Spaces
Let be a family of ⊕-sb-metrics spaces. Further, assume that is an ⊕-sbki-metric space. Because the family is finite, we may take . Subsequently, all are ⊕-sbk-metrics. We define and by where and are i-th coordinates of x and y, respectively.
Theorem 5. is the product of the family in the category ⊕-SbMetr , where ⊕ is a distributive continuous on the bottom extended t-conorm. Besides, the topology that is induced by the ⊕-sb-metric d on X coincides with the product of the topologies induced by ⊕-sb-metrics .
Proof. First, we have to show that
d is an ⊕-sb-metric. Properties
and
of
d follow obviously from the corresponding properties of all
, respectively. To show the validity of axiom
, we fix
, and applying the axiom
for every
and disributivity of ⊕, we are reasoning, as follows:
Further, let
be an ⊕-sb-metric space and let for every
a mapping
be continuous. Subsequently, from the construction, it is clear that, by setting
, we obtain a continuous mapping
, such that
where
is the projection. Thus,
is the product of the family
in the category ⊕
-SbMetr. To show the second statement of the theorem, notice first that all of the projections
are continuous in ⊕
-SbMetr. Indeed, let
and a point
be given. Because ⊕ is monotone and 0 is the neutral element of ⊕, we have
for any
. Therefore,
for any
whenever
. Hence, the topology induced by the ⊕-sb-metric
d is stronger or equal than the topology of the product of topologies
induced by ⊕-sb-metrics
.
To prove the converse inequality, we show that every set
that is open in
is also open in the product topology
. Let
, we fix
such that
Subsequently, for each coordinate
i, take the ball
. Applying distributivity and monotonicity of the t-conorm ⊕, we conclude that
for all
implies
. However, this means that
and hence
U is open in the product topology. □
In case all in the previous theorem are ⊕-sbk-metrics with a fixed k, we get the following corollary:
Corollary 2. defined in the previous theorem is the product of the family in the category⊕-SbkMetr.
Remark 9. Patterned after the above proof it can also be proved in the context of ⊕-metric for any distributive continuous on the bottom extended t-conorm. Certainly, in case this result can be found in almost any book on topology or functional analysis. Concerning the category of ⊕-b-metric spaces, we have the following:
Let be a family of ⊕-b-metrics. We define and by where and are i-th coordinates of x and y, respectively. Subsequently, patterned after the proof of Theorem 5, one can show that is the product of the family in the category ⊕-bMetr. However, for the reasons discussed above, we cannot confirm the topological part of the previous theorem.
6.2. Products of Infinite Families of ⊕-Sb-Metrics
In case of infinite products, we have to restrict our results to the cases when
and
. Patterned after the standard proof of the theorem about the product of a countable family of ordinary metrics, see e.g., [
19], we have to replace a given ⊕-sb-metric by an equivalent bounded ⊕-sb-metric, which is a metric that induces the same topology on the underlying set
X. In order to do this, we ask the additional assumption that ⊕ is compressible, which is
Lemma 2. Operations and are compressible.
Proof. Straightforward. □
We consider the cases and separately. First, we consider the case , which is ⊕-sb-metrics are “ordinary” sb-metrics.
Theorem 6. Let be an sbk-metric. Subsequently, by setting a bounded sbk-metric is defined. Besides, sbk-metrics d and ρ are equivalent.
Proof. Obviously,
satisfies properties
and
. To show that axiom
holds for
d, we refer to the compressibility of
and the validity of this axiom for
and have
The equivalence of the ⊕-sbk-metrics
d and
is obvious, since they have the same convergent sequences. □
Let
be a countable family of sbk-metric spaces. Let
. We define
and
by
where
and
are
coordinates of
x and
y, respectively. Further, let
be the corresponding projection.
Theorem 7. is the product of the family inSbkMetr. The topology that is induced by ⊕-sbk-metric d coincides with the topology of the product of the topologies .
Proof. We are reasoning patterned after the proof of Theorem 4.2.2 in [
19]. Since
for every
,
and hence the definition of the sum is correct. To show that
d is thus defined an sbk-metric, note that the validity of axioms
and
is obvious. We verify the validity of axiom
, as follows. Because all
are sbk-metrics (with the same k) for the points
, we have:
Multiplying these inequalities by
and summing them up, we obtain
Further, let be a sbk-metric space and let for every a mapping be continuous. Subsequently, from the construction, it is clear that, by setting , we obtain a continuous mapping , such that where is the projection. Thus, is indeed the product of the spaces in the category SbkMetr.
The proof that topology
induced by sbk-metric
d coincides with the product of the topologies
induced by sbk-metrics
can be done repeating verbatim the proof of Theorem 4.2.2 in [
19]. □
Corollary 3. is the product of the family inSbkMtrz.
Let now
be a countable family of ∨-sbk-metric spaces and let
. We define
and
by
where
and
are
coordinates of
x and
y, respectively.
Theorem 8. is the product of the family in ∨-SbkMetr. The topology induced by ∨-sbk-metric d coincides with the product of the topologies induced by ∨-sbk-metrics . Hence is also the product of in ∨-SbkMtrz.
Proof. Because all , definition of d is correct and The properties (reflexivity) and (symmetry) of d are obvious. We prove the property for d, as follows.
Let
, then
Further, let be an ∨-sbk-metric space and let for every a mapping be continuous. Subsequently, from the construction it is clear that by setting we obtain a continuous mapping such that where are the projections. Hence, the defined construction is indeed a product in the category of bounded ∨-sbk-metric spaces.
To see that the topology
induced by
d on the product
coincides with the product of topologies
, just notice that, as a base for the topology
, we may take the family of all balls
□
Corollary 4. is the product of the family in ∨-SbkMtrz.
Remark 10. The analogous statement of Theorems 5 and 7 in the category of metric spaces is a classic result that can be found in almost every textbook in functional analysis and topology. Patterned after the proof of Theorem 5, analogous result for ⊕-metric spaces can also be proved in the case of any distributive continuous at the bottom extended t-conorm.
Unfortunately, for the reasons discussed above we have no any results about products in categories of ⊕-b-metric spaces⊕-b-MetrSand⊕-b-MetrT.
6.3. Co-Products (Direct Sums) of Families of ⊕-Sbk-Metrics
Speaking about co-products (or direct sums) of ⊕-sbk-metrics, we do not distinguish between finite and infinite cases, because, in both cases, we need to assume that the ⊕-sbk-metrics are bounded. The only difference between the finite and infinite cases is that in the infinite case we have to assume that the parameter k in all ⊕-sbk-metrics is the same, while, in the finite case, we may take k as the largest of all . Accordingly, in the proof we assume that k is the same for all .
Let be a family of ⊕-sbk-metric spaces and let . We define (that is X is the disjoint union of the sets and by if and otherwise.
Theorem 9. is an ⊕-sbk-metric and is the coproduct of the family in the category ⊕-SbkMetr, where ⊕ is a compressible continuous on the bottom extended t-conorm. The topology that is induced by d coincides with the topology of the coproduct (direct sum) of the spaces .
Proof. We first show that d is an ⊕-sbk-metric. The validity of properties and for d is obvious. To verify , let and consider the three cases.
- (1)
If there exists such that , then the conclusion follows, since is an ⊕-sbk-metric.
- (2)
If but , then , but . For example, in case we have
- (3)
If , then while .
From the construction it is clear that the inclusion mappings are continuous and that the topology that is induced by d coincides with the coproduct (direct sum) of the topologies induced by .
Finally, let be an ⊕-sbk-metric space and let for every a mapping be continuous. Subsequently, from the construction it is clear that by setting if , we obtain a continuous mapping , such that . □
Corollary 5. is the coproduct of the family in the category ⊕-SbkMtrz, where ⊕ is a compressible continuous on the bottom extended t-conorm.
7. Conclusions
In the paper, we have used the concept of an extended t-conorm ⊕ in order to define ⊕-metric, ⊕-b-metric and ⊕-sb-metric spaces. In case when ⊕ is the Łukasiewicz t-norm extended from the triangle
, they become “classic” metrics, b-metrics, and sb-metrics, respectively. The structure of these spaces was discussed and the property of continuity for mappings of such spaces was studied. ⊕-metric, ⊕-b-metric, and ⊕-sb-metric spaces, and their continuous mappings allow us to speak about several categories. We study some properties of these categories specifically the existence of products and coproducts in these categories. Throughout the paper our main interest is in ⊕-sb-metric spaces, in particular in case of sb-metric spaces. Answering a question post in [
1], we constructed a series of sb-metric spaces that fail to be metric.
Concerning the plans for the future work, we see the following:
To study completeness and Baire property for ⊕-sb-metric spaces.
To investigate topological properties of sb-metric spaces. It is clear that they are Hausdorff. However, are they regular, completely regular, normal? What additional properties can be proved for separable sb-metric spaces?
Characterize compact subsets in sb-metric spaces.
We plan to study further categorical properties of sb-metric spaces and sb-metrizable spaces
To study the relations between the categories of metric-type spaces.
Consider the perspective for studying ⊕-metric-type structures for a general extended t-conorm. By now we have the full bodied theory only in case ⊕ is distributive and continuous. Unfortunately, we only have two examples of extended t-norms with such properties: and . Are there other distributive continuous extended t-norms (except of obvious modification of the previous two)?