# On Metric-Type Spaces Based on Extended T-Conorms

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## Abstract

**:**

## 1. Introduction

## 2. Extended T-Conorms

**Definition**

**1.**

- (⊕
_{1}) ⊕ is commutative, that is $\alpha \oplus \beta =\beta \oplus \alpha ;$ - (⊕
_{2})⊕ is associative, that is $\alpha \oplus (\beta \oplus \gamma )=(\alpha \oplus \beta )\oplus \gamma ;$ - (⊕
_{3}) ⊕ is monotone, that is $\alpha \le \beta \u27f9\alpha \oplus \gamma \le \beta \oplus \gamma ;$ - (⊕
_{4}) 0 is the neutral element for ⊕, that is $\alpha \oplus 0=\alpha .$

**Remark**

**1.**

**Definition**

**2.**

- (⊕
_{5}) $k\xb7(\alpha \oplus \beta )=k\xb7\alpha \oplus k\xb7\beta $. - ⊕ is called compressible, if,
- (⊕
_{6}) $\alpha \le \beta \oplus \gamma \phantom{\rule{4pt}{0ex}}\u27f9\frac{\alpha}{\alpha +1}\le \frac{\beta}{\beta +1}\oplus \frac{\gamma}{\gamma +1}.$ - ⊕ is called continuous at the bottom level, if this operation is continuous in all points of $\left\{0\right\}\times {\mathbb{R}}^{+}\subset {\mathbb{R}}^{+}\times {\mathbb{R}}^{+}$ (and, hence, by symmetry also on ${\mathbb{R}}^{+}\times \left\{0\right\})$.

**Example**

**1.**

**Example**

**2.**

**Lemma**

**1.**

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

## 3. Metric-Type Structures Based on Extended T-Conorms

**Definition**

**3.**

- (m
_{1}) $d(x,x)=0\phantom{\rule{4pt}{0ex}}\forall \phantom{\rule{4pt}{0ex}}x\in X;$ - (m
_{2}) $d(x,y)=d(y,x)\phantom{\rule{4pt}{0ex}}\forall \phantom{\rule{4pt}{0ex}}x,y\in X;$ - A semi-metric is a semi pseudometric satisfying the following strengthen version of the first axiom:
- (m
_{1’}) $d(x,y)=0\u27fax=y.$

**Definition**

**4.**

- (m
_{3}) ⊕-based (pseudo)metric, or just ⊕-(pseudo)metric, if- $d(x,y)\le d(x,z)\oplus d(z,y)\forall x,y,z\in X;$

- (mb
_{3}) ⊕-based b-(pseudo)metric, or just ⊕-b-(pseudo)metric if there exists $k\ge 1$ such that- $d(x,y)\le k\xb7\left(d\right(x,z)\oplus d(z,y\left)\right)\phantom{\rule{4pt}{0ex}}\forall \phantom{\rule{4pt}{0ex}}x,y,z\in X;$

- (msb
_{3}) ⊕-based sb-(pseudo)metric or just ⊕-sb-(pseudo)metric, if there exists $k\ge 1$, such that- $d(x,y)\le d(x,z)\oplus k\xb7d(z,y)\phantom{\rule{4pt}{0ex}}\forall \phantom{\rule{4pt}{0ex}}x,y,z\in X$.

**Remark**

**2.**

## 4. Categories of ⊕-Metric-Type Spaces

**Definition**

**5.**

**Theorem**

**1.**

**Proof.**

**Remark**

**3.**

**Corollary**

**1.**

**Theorem**

**2.**

**Proof.**

**Definition**

**6.**

**Remark**

**4.**

**Definition**

**7.**

**Theorem**

**3.**

- 1.
- $f:(X,{d}_{X})\to (Y,{d}_{Y})$ is continuous;
- 2.
- for every $a\in X$, every $\epsilon >0$ there exists $\delta >0$ such that ${d}_{Y}(f\left(a\right),f\left(x\right))<\epsilon $ whenever ${d}_{X}(a,x)<\delta ;$
- 3.
- if a sequence ${x}_{1},{x}_{2},\dots {x}_{n}\dots $ converges to a point ${x}_{0}$ in the space $(X,{d}_{X})$, then the sequence $f\left({x}_{1}\right),f\left({x}_{2}\right),\dots f\left({x}_{n}\right)\dots $ converges to the point $f\left({x}_{0}\right)$ in the space $(Y,{d}_{Y})$.

**Theorem**

**4.**

**Proof.**

- The objects of the category
**⊕-Metr**are pairs $(X,d)$ where X is a set and d is an ⊕-metric on it. The morphisms of the category**⊕-Metr**are continuous mappings $f:(X,{d}_{X})\to (Y,{d}_{Y})$. - The objects of the category
**⊕-Mtrz**are pairs $(X,{\mathcal{T}}_{d})$ where X is a set and ${\mathcal{T}}_{d}$ is a topology induced by some ⊕-metric d. The morphisms of the category**⊕-Mtrz**are continuous mappings $f:(X,{\mathcal{T}}_{{d}_{X}})\to (Y,{\mathcal{T}}_{{d}_{Y}})$. - The objects of the category
**⊕-SbMetr**of ⊕-sb-metric spaces are pairs $(X,d)$ where X is a set and d is an ⊕-sb-metric on it. The morphisms of the category**⊕-SbMetr**are continuous mappings $f:(X,{d}_{X})\to (Y,{d}_{Y})$.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 $(X,{\mathcal{T}}_{d})$, where X is a set and ${\mathcal{T}}_{d}$ is a topology that is induced by some ⊕-sb-metric d. The morphisms of the category**⊕-SbMtrz**are continuous mappings $f:(X,{\mathcal{T}}_{{d}_{X}})\to (Y,{\mathcal{T}}_{{d}_{Y}})$. 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 ${\mathcal{S}}_{d}$ be the family of all unions of open balls, that is$${\mathcal{S}}_{d}=\{U\subseteq X:\exists {B}_{d}({a}_{i},{\epsilon}_{i}),i\in I\mathrm{such}\mathrm{that}U={\bigcup}_{i\in I}{B}_{d}({a}_{i},{\epsilon}_{i})\}.$$The family ${\mathcal{S}}_{d}$ is obviously a supratopology (see e.g., [16,17]), that is closed under taking arbitrary unions. ${\mathcal{S}}_{d}$ need not be a topology: the intersection of even two elements ${U}_{1},{U}_{2}\in {\mathcal{S}}_{d}$ need not be in ${\mathcal{S}}_{d}$ since an open ball need not be open in ${\mathcal{S}}_{d}$. Let ⊕
**-b-MetrS**be a category whose objects are pairs $(X,d)$ where X is a set and d is an ⊕-b-metric on it and whose morphisms are continuous mappings $f:(X,{\mathcal{S}}_{{d}_{X}})\to (Y,{\mathcal{S}}_{{d}_{Y}})$ - (b)
- (see [3]) We call a set $U\subseteq X$${\mathcal{T}}_{d}$-open if for every $x\in X$ there exists $\epsilon >0$, such that $B(a,\epsilon )\subseteq U.$ One can easily notice that ${U}_{1},{U}_{2}\in {\mathcal{T}}_{d}\u27f9{U}_{1}\cap {U}_{2}\in {\mathcal{T}}_{d}$ and the union of any family of ${\mathcal{T}}_{d}$-open sets is ${\mathcal{T}}_{d}$-open. Thus ${\mathcal{T}}_{d}$ is indeed a topology on X.Obviously, each $U\in {\mathcal{T}}_{d}$ can be expressed as a union of some open balls and for this reason it belongs to ${\mathcal{S}}_{d}$. Thus, ${\mathcal{T}}_{d}\subseteq {\mathcal{S}}_{d}.$ On the other hand not every open ball $B(a,\epsilon )$ needs to be ${\mathcal{T}}_{d}$-open and, hence, generally ${\mathcal{S}}_{d}\ne {\mathcal{T}}_{d}.$ Let ⊕
**-b-MetrT**be the category whose objects are pairs $(X,d)$ where X is a set and d is an ⊕-b-metric on it and whose morphisms are continuous mappings $f:(X,{\mathcal{T}}_{{d}_{X}})\to (Y,{\mathcal{T}}_{{d}_{Y}})$.

## 5. Examples

#### 5.1. Examples of Sb-Metric Spaces

**Example**

**6.**

**Proof.**

**Remark**

**5.**

**Example**

**7.**

**Proof.**

**Remark**

**6.**

**Example**

**8.**

**Proof.**

**Remark**

**7.**

#### 5.2. Examples of ⊕-Metric Type Spaces

**Example**

**9.**

- (1)
- If $\oplus =+$,then d is an ⊕-sb2-metric (simply sb2-metric).
- (2)
- If $\oplus =\vee $ then d is an ⊕-sb4-metric (might be called ultra sb4-metric).
- (3)
- If $\oplus ={\oplus}_{T}$ (see Example 2.7) then d is an ⊕-metric. Note that ${\oplus}_{T}$ 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 $h\ge 2$, then d is an ⊕-metric. If $h<2$, then d is an ⊕-sb- $\frac{4-h}{2}$-metric.

**Example**

**10.**

- (a)
- If $k=1$, then d is an ⊕-metric.
- (b)
- If $k<1$, then d is an ⊕-$b\frac{1}{k}$-metric.
- (c)
- If $k>1$, then d is an ⊕-$bk$-metric.

**Proof.**

**Example**

**11.**

- (a)
- If $k=1$, then d is an ⊕-metric.
- (b)
- If $k<1$ and $k+h\ge 1$, then d is an ⊕-metric.
- (c)
- If $k<1$ and $k+h<1$, then d is an ⊕-$b\frac{1}{k}$-metric.
- (d)
- If $k>1$ and $1+h\ge k$, then d is an ⊕-metric.
- (e)
- If $k>1$ and $1+h<k$, then d is an ⊕-$b\frac{k}{1+h}$ metric.

**Proof.**

**Remark**

**8.**

**Example**

**12.**

## 6. Products of ⊕-Sb-Metric Spaces

#### 6.1. Products of Finite Families of ⊕-Sb-Metric Spaces

_{i}-metric space. Because the family is finite, we may take $k=max\{{k}_{1},\dots ,{k}_{n}\}$. Subsequently, all ${d}_{i}$ are ⊕-sbk-metrics. We define $X={\prod}_{i=1}^{n}{X}_{i}$ and $d:X\times X\to [0,\infty )$ by $d(x,y)={\u2a01}_{i=1}^{n}d({x}_{i},{y}_{i})$ where $x,y\in X$ and ${x}_{i},{y}_{i}$ are i-th coordinates of x and y, respectively.

**Theorem**

**5.**

**-SbMetr**, where ⊕ is a distributive continuous on the bottom extended t-conorm. Besides, the topology ${\mathcal{T}}_{d}$ that is induced by the ⊕-sb-metric d on X coincides with the product of the topologies ${\mathcal{T}}_{{d}_{i}}$ induced by ⊕-sb-metrics ${d}_{i}$.

**Proof.**

**-SbMetr**. To show the second statement of the theorem, notice first that all of the projections ${p}_{i}:(X,d)\to ({X}_{i},{d}_{i})$ are continuous in ⊕

**-SbMetr**. Indeed, let $\epsilon >0$ and a point $a\in {\prod}_{i}{X}_{i}$ be given. Because ⊕ is monotone and 0 is the neutral element of ⊕, we have $a,b\le a\oplus b$ for any $a,b\in [0,1]$. Therefore, ${d}_{i}({a}_{i},{x}_{i})<\epsilon $ for any $i=1,\dots n$ whenever $d(a,x)<\epsilon $. Hence, the topology induced by the ⊕-sb-metric d is stronger or equal than the topology of the product of topologies ${\mathcal{T}}_{{d}_{i}}$ induced by ⊕-sb-metrics ${d}_{i}$.

**Corollary**

**2.**

**-SbkMetr**.

**Remark**

**9.**

**-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

- (⊕
_{6}) $a\le b\oplus c\phantom{\rule{4pt}{0ex}}\u27f9\frac{a}{1+a}\le \frac{b}{1+b}\oplus \frac{c}{1+c}$

**Lemma**

**2.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**SbkMetr**. The topology that is induced by ⊕-sbk-metric d coincides with the topology of the product of the topologies ${\mathcal{T}}_{{d}_{i}}$.

**Proof.**

**SbkMetr**.

**Corollary**

**3.**

**SbkMtrz**.

**Theorem**

**8.**

**-SbkMetr**. The topology ${\mathcal{T}}_{d}$ induced by ∨-sbk-metric d coincides with the product of the topologies ${\mathcal{T}}_{{d}_{i}}$ induced by ∨-sbk-metrics ${d}_{i}$. Hence $(X,d)$ is also the product of $\{({X}_{i},{d}_{i}):i\in \mathbb{N}\}$ in ∨

**-SbkMtrz**.

**Proof.**

**Corollary**

**4.**

**-SbkMtrz**.

**Remark**

**10.**

**-b-MetrS**and⊕

**-b-MetrT**.

#### 6.3. Co-Products (Direct Sums) of Families of ⊕-Sbk-Metrics

**Theorem**

**9.**

**-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 $({X}_{i},{d}_{i})$.

**Proof.**

- (1)
- If there exists $i\in I$ such that $x,y,z\in {X}_{i}$, then the conclusion follows, since ${d}_{i}$ is an ⊕-sbk-metric.
- (2)
- If $x,y\in {X}_{i},$ but $z\notin {X}_{i}$, then $d(x,y)<1$, but $d(x,z)\oplus d(z,y)\ge 1$. For example, in case $\oplus =+$ we have $d(x,z)+d(z,y)=2,d(x,z)\vee d(z,y)=1.$
- (3)
- If $x\in {X}_{i},y\in {X}_{j},i\ne j$, then $d(x,y)=1$ while $d(x,z)\oplus d(z,y)>1$.

**Corollary**

**5.**

**-SbkMtrz**, where ⊕ is a compressible continuous on the bottom extended t-conorm.

## 7. Conclusions

- 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: $\otimes =+$ and $\otimes =\vee $. Are there other distributive continuous extended t-norms (except of obvious modification of the previous two)?

## Author Contributions

## Funding

## Conflicts of Interest

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Öner, T.; Šostak, A. On Metric-Type Spaces Based on Extended *T*-Conorms. *Mathematics* **2020**, *8*, 1097.
https://doi.org/10.3390/math8071097

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Öner T, Šostak A. On Metric-Type Spaces Based on Extended *T*-Conorms. *Mathematics*. 2020; 8(7):1097.
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**Chicago/Turabian Style**

Öner, Tarkan, and Alexander Šostak. 2020. "On Metric-Type Spaces Based on Extended *T*-Conorms" *Mathematics* 8, no. 7: 1097.
https://doi.org/10.3390/math8071097