# Topology in the Alternative Set Theory and Rough Sets via Fuzzy Type Theory

## Abstract

**:**

## 1. Introduction

- Using formalism of FTT, we will unify rough set and fuzzy rough set theories into one formal system. Their concepts can be distinguished only semantically in a model.
- Using formalism of FTT, we will show the equivalence of the concepts of rough set theory with some of the topological concepts introduced earlier in AST. All the considered concepts are then passed into the fuzzy set theory by introducing a proper model.
- We will let the readers know about a very interesting set theory that claims to become an alternative to the classical one, and that has the potential to stand behind foundations of new mathematics.

## 2. Few Selected Concepts of AST

**Definition**

**1.**

- (a)
- A figure is a class X fulfilling the following condition:$$(\forall x)(\forall y)(x\in X\&(x\doteq y)\Rightarrow y\in X).$$If X is a class then the figure of X is the class$$\mathrm{Fig}\left(X\right)=\{x\mid (\exists y\left)\right(y\in X\&(y\doteq x)\left)\right\}.$$
- (b)
- A monad of an element x is a class$$\mathrm{Mon}\left(x\right)=\{y\mid y\doteq x\}=\mathrm{Fig}\left(\right\{x\left\}\right).$$
- (c)
- Classes$X,Y$are separable,$\mathrm{Sep}(X,Y)$, if there is a set-theoretically definable class Z such that$\mathrm{Fig}\left(X\right)\subseteq Z$and$Z\cap \mathrm{Fig}\left(Y\right)=\varnothing $.
- (d)
- Closure of a class X is the class$$\mathrm{Clo}\left(X\right)=\{x\mid \neg \mathrm{Sep}(\left\{x\right\},X\left)\right\}.$$

## 3. Basic Concepts of Rough Set Theory

**Definition**

**2.**

## 4. Brief Overview of FTT

_{Δ}-algebra of truth values (cf. [4]) (Let us remark that we can also introduce a core fuzzy type theory on the basis of which all other kinds of FTT can be obtained—see [7]).

#### 4.1. Truth Degrees

_{Δ}-algebra. The latter is the algebra

- (i)
- $\langle E,\vee ,\wedge ,\mathbf{0},\mathbf{1}\rangle $ is a bounded lattice (
**0**,**1**, are the least and the greatest elements, respectively); - (ii)
- $\langle L,\otimes ,\mathbf{1}\rangle $ is a commutative monoid;
- (iii)
- The operation → is a residuation operation with respect to ⊗, i.e.$$a\otimes b\le c\hspace{1em}\mathrm{iff}\hspace{1em}a\le b\to c;$$
- (iv)
- $(a\to b)\vee (b\to a)=\mathbf{1}$; (prelinearity)
- (v)
- $(a\to b)\to b=a\vee b$.

_{Δ}-algebra for which $E=[0,1]$ and the operations are defined as follows:

#### 4.2. Fuzzy Equality

- (i)
- reflexivity$$\doteq \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}(m,m)=1;$$
- (ii)
- symmetry$$\doteq \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}(m,{m}^{\prime})=\phantom{\rule{0.277778em}{0ex}}\doteq \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}({m}^{\prime},m);$$
- (iii)
- ⊗-transitivity$$\doteq \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}(m,{m}^{\prime})\phantom{\rule{0.277778em}{0ex}}\otimes \phantom{\rule{0.277778em}{0ex}}\doteq \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}({m}^{\prime},{m}^{\u2033})\le \phantom{\rule{0.277778em}{0ex}}\doteq \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}(m,{m}^{\u2033}).$$

#### 4.3. Syntax

_{α}, the set of all formulas by Form. Interpretation of a formula ${A}_{\beta \alpha}$ is a function from the set of objects of type $\alpha $ into the set of objects of type $\beta $. Thus, if $B\in {Form}_{\beta \alpha}$ and $A\in {Form}_{\alpha}$ then $\left(BA\right)\in {Form}_{\beta}$. Similarly, if $A\in {Form}_{\beta}$ and ${x}_{\alpha}\in J$, $\alpha \in $Types is a variable and then $\lambda {x}_{\alpha}\phantom{\rule{0.166667em}{0ex}}{A}_{\beta}\in {Form}_{\beta \alpha}$ is a formula whose interpretation is a function which assigns to each object of type $\alpha $ an object of type $\beta $ represented by the formula ${A}_{\beta}$.

**Remark**

**1.**

- (i)
- ${\equiv}_{\left(oo\right)o}\text{}:=\phantom{\rule{0.166667em}{0ex}}\lambda {x}_{o}\lambda {y}_{o}\left({\mathbf{E}}_{\left(oo\right)o}\phantom{\rule{0.166667em}{0ex}}{y}_{o}\right){x}_{o}$,
- (ii)
- ${\equiv}_{\left(o\u03f5\right)\u03f5}\phantom{\rule{0.166667em}{0ex}}\text{}:=\lambda {x}_{\u03f5}\lambda {y}_{\u03f5}\left({\mathbf{E}}_{\left(oo\right)o}\phantom{\rule{0.166667em}{0ex}}{y}_{\u03f5}\right){x}_{\u03f5}$,
- (iii)
- ${\equiv}_{\left(o\right(\beta \alpha \left)\right)\left(\beta \alpha \right)}\phantom{\rule{0.166667em}{0ex}}\text{}:=\lambda {f}_{\beta \alpha}\lambda {g}_{\beta \alpha}\left({\mathbf{E}}_{\left(o\right(\beta \alpha \left)\right)\left(\beta \alpha \right)}\phantom{\rule{0.222222em}{0ex}}{g}_{\beta \alpha}\right){f}_{\beta \alpha}$.

**Remark**

**2.**

- Rule (R): From ${A}_{\alpha}\equiv {A}_{\alpha}^{\prime}$ and $B\in {Form}_{o}$ infer ${B}^{\prime}$, where ${B}^{\prime}$ comes from B by replacing one occurrence of ${A}_{\alpha}$, which is not preceded by $\lambda $, by ${A}_{\alpha}^{\prime}$.
- Rule (N): from ${A}_{o}$ infer $\Delta {A}_{o}$.

#### 4.4. Semantics

- (i)
- The ${\epsilon}_{\Delta}$ is a linearly ordered MV
_{Δ}algebra of truth degrees. We put ${M}_{o}=E$; ${M}_{\u03f5}$ is a set of arbitrary elements. - (ii)
- ${\doteq}_{\alpha}:{M}_{\alpha}\times {M}_{\alpha}\to E$ is a separated fuzzy equality on ${M}_{\alpha}$ introduced in Section 4.2.As a special case, we define: ${\doteq}_{o}:=\leftrightarrow $ (biresiduation). The fuzzy equality ${\doteq}_{\u03f5}$ between elements of type $\u03f5$ must be given explicitly. The fuzzy equality between functions $f,{f}^{\prime}\in {M}_{\beta \alpha}$ is defined in (11).
- (iii)
- If $\alpha =\gamma \beta \in Types$ then ${M}_{\gamma \beta}\subseteq {M}_{\gamma}^{{M}_{\beta}}$. As a special case, the set ${M}_{oo}\cup {M}_{\left(oo\right)o}$ contains all the operations of the algebra ${\epsilon}_{\Delta}$.

**Theorem**

**1.**

- (a)
- A theory T is consistent iff it has a general model$\mathcal{M}$.
- (b)
- For every theory T and a formula${A}_{o}$$$T\u22a2{A}_{o}\hspace{1em}iff\hspace{1em}T\vDash {A}_{o}.$$

**1**iff ${\mathcal{M}}_{p}\left({A}_{o}\right)\in (\mathbf{0},\mathbf{1})$ and ${\mathcal{M}}_{p}\left(\mathbf{Y}{A}_{o}\right)=$

**1**iff ${\mathcal{M}}_{p}\left({A}_{o}\right)>0$ holds in any model $\mathcal{M}$. The formulas $\mathbf{Y}{A}_{o}$ and $\widehat{\mathbf{Y}}{A}_{o}$ are crisp.

**Lemma**

**1.**

- (a)
- $\u22a2(\exists {z}_{o})({z}_{o}\equiv {y}_{o})$,
- (b)
- $\u22a2{y}_{o}\equiv (\exists {z}_{o})({z}_{o}\&({z}_{o}\equiv {y}_{o}))$,
- (c)
- $\u22a2{y}_{o}\equiv (\exists {z}_{o})({z}_{o}\&\Delta ({z}_{o}\equiv {y}_{o}))$,
- (d)
- $\u22a2(\exists {z}_{o}){z}_{o}$,
- (e)
- If${\mathbf{r}}_{o}$is a constant then$\u22a2(\forall {x}_{\alpha}){\mathbf{r}}_{o}\equiv {\mathbf{r}}_{o}$and$\u22a2(\exists {x}_{\alpha}){\mathbf{r}}_{o}\equiv {\mathbf{r}}_{o}$.
- (f)
- $\u22a2({x}_{o}\equiv {y}_{o})\equiv (\exists {z}_{o})(({x}_{o}\equiv {z}_{o})\&({z}_{o}\equiv {y}_{o}))$,
- (g)
- $\u22a2{x}_{o}\Rightarrow \mathbf{Y}{x}_{o}$.

**Proof.**

## 5. Rough Sets and AST via Fuzzy Type Theory

#### 5.1. Fuzzy Set Theory in FTT

**Lemma**

**2.**

- (a)
- $\u22a2\mathrm{Crisp}{\varnothing}_{o\alpha}$and$\u22a2\mathrm{Crisp}{\mathbf{V}}_{o\alpha}$.
- (b)
- $\u22a2\mathrm{Crisp}\left(\mathrm{Supp}\text{}X\right)$.
- (c)
- $\u22a2X\subseteq \mathrm{Supp}\text{}X$.
- (d)

**Proof.**

**Lemma**

**3.**

- (a)
- $\u22a2\mathrm{Crisp}\left(\mathrm{sg}{u}_{\alpha}\right)$.
- (b)
- $\u22a2\mathrm{Supp}\left(\mathrm{sg}u\right)\equiv \mathrm{sg}u$.

**Proof.**

- (a)
- is obvious.
- (b)
- This follows from the following sequence of provable formulas:$\u22a2\mathrm{Supp}\left(\mathrm{sg}u\right)\equiv \lambda {v}_{\alpha}\xb7\mathrm{Y}\left(\mathrm{sg}u\right)v$, $\u22a2\mathrm{Supp}\left(\mathrm{sg}u\right)\equiv \lambda {v}_{\alpha}\xb7\mathrm{Y}(\Delta (u\approx v))$ and$\u22a2\mathrm{Supp}\left(\mathrm{sg}u\right)\equiv \mathrm{sg}u$.

#### 5.2. Transfer of Selected Concepts of AST into Fuzzy Set Theory

- (EV1)
- $t\approx t$,
- (EV2)
- $t\approx u\equiv u\approx t$,
- (EV3)
- $t\approx u\&u\approx z\xb7\Rightarrow t\approx z$,
- (EV4)
- $\Delta (z\equiv t)\equiv \Delta (z\approx t)$.

**Definition**

**3.**

- (i)
- A figure of a fuzzy set ${x}_{o\alpha}$ is defined by$${\mathrm{Fig}}_{\left(o\alpha \right)\left(o\alpha \right)}\equiv \lambda {x}_{o\alpha}\lambda {u}_{\alpha}\xb7(\exists {v}_{\alpha})(xv\&(u\approx v)).$$
- (ii)
- A monad of an element u is defined by$${\mathrm{Mon}}_{\left(o\alpha \right)\alpha}\equiv \lambda {u}_{\alpha}\lambda {v}_{\alpha}\xb7u\approx v.$$
- (iii)
- A property characterizing a fuzzy set to be a figure is represented by the formula$${\mathbf{FIG}}_{o\left(o\alpha \right)}\equiv {\lambda x}_{o\alpha}\xb7(\forall {u}_{\alpha})(\forall {v}_{\alpha})(xu\&(u\approx v)\Rightarrow xv).$$

**Lemma**

**4.**

- (a)
- $\u22a2\mathbf{FIG}{\varnothing}_{\alpha}$ and $\u22a2\mathbf{FIG}{\mathbf{V}}_{\alpha}$.
- (b)
- $\u22a2X\subseteq \mathrm{Fig}X$.
- (c)
- $\u22a2\mathbf{FIG}\left(\mathrm{Fig}X\right)$, i.e.,$\mathrm{Fig}X$is a figure.
- (d)
- $\u22a2\mathrm{Fig}\left(\mathrm{sg}u\right)\equiv \mathrm{Mon}u$.
- (e)
- If$\u22a2X\subseteq Y$then$\u22a2\mathrm{Fig}X\subseteq \mathrm{Fig}Y$.
- (f)
- $\u22a2\mathrm{Fig}(X\cup Y)\equiv (\mathrm{Fig}X\cup \mathrm{Fig}Y)$.

**Proof.**

**Lemma**

**5.**

- (a)
- $\u22a2\mathbf{FIG}X$.
- (b)
- $\u22a2X\equiv \lambda {v}_{\alpha}\xb7(\exists u)((u\approx v)\&Xu)$.
- (c)
- $\u22a2(\forall {u}_{\alpha})(Xu\Rightarrow (\mathrm{Mon}u\subseteq X))$.

**Proof.**

**Lemma**

**6.**

- (a)
- $\u22a2\mathbf{FIG}\left(X\right)$iff$\u22a2X\equiv \mathrm{Fig}X$.
- (b)
- $\u22a2\mathrm{Fig}\left(\mathrm{Fig}X\right)\equiv \mathrm{Fig}X$.

**Proof.**

**Lemma**

**7.**

**Proof.**

**Definition**

**4.**

- (i)
- Separability of two fuzzy sets is characterized by the formula$${\mathrm{Sep}}_{o\left(o\alpha \right)\left(o\alpha \right)}\equiv \lambda {x}_{o\alpha}\lambda {y}_{o\alpha}\xb7(\forall {v}_{\alpha})(\forall {v}_{\alpha})(\left(\mathrm{Fig}x\right)v\wedge \left(\mathrm{Fig}y\right)w\Rightarrow \neg \mathrm{Y}(v\approx w)).$$
- (ii)
- $X,Y$are separable if$\u22a2\left(\mathrm{Sep}X\right)Y$.
- (iii)
- A special case is separability of an element u from X:$${\mathrm{Sep}}_{o\alpha \left(o\alpha \right)}\equiv \lambda {x}_{o\alpha}\lambda {u}_{\alpha}\xb7(\forall {v}_{\alpha})(\forall {w}_{\alpha})(\left(\mathrm{Fig}X\right)v\wedge \left(\mathrm{Mon}u\right)w\Rightarrow \neg \mathrm{Y}(v\approx w)).$$
- (iv)
- Closure of a fuzzy set X is a fuzzy set$${\mathrm{Clo}}_{(}o\alpha )\left(o\alpha \right)X\equiv \lambda {u}_{\alpha}\xb7\neg \left(\mathrm{Sep}X\right)u.$$

**Remark**

**3.**

- (a)
- Formula (21) means that if fuzzy sets${X}_{o\alpha},{Y}_{o\alpha}$are separated, then if u belongs to$\mathrm{Fig}X$in a non-zero degree and v belongs to$\mathrm{Fig}Y$in a non-zero degree then they cannot be equal in a non-zero degree. Interpretation of this formula, however, can be many-valued, i.e., we can have two fuzzy sets separable only by some degree. Full separability is obtained if provability of$\left(\mathrm{Sep}X\right)Y$is assured—cf. item (ii).
- (b)
- Formula (22) is, in fact, different from Formula (21). The special case of the latter is$\left(\mathrm{Sep}X\right)\left(\mathrm{sg}u\right)$. For obvious reasons, however, we will use the same symbol both for separability of two fuzzy sets and separability of an element from a fuzzy set, if no misunderstanding can occur.

- (c)
- Closure of X is a fuzzy set of elements${u}_{\alpha}$, to which there are elements${v}_{\alpha}$from the figure$\mathrm{Fig}{X}_{o\alpha}$, and an element${w}_{\alpha}$from the monad of$\mathrm{Mon}u$that are fuzzy equal in a non-zero degree.

**Lemma**

**8.**

**Lemma**

**9.**

- (a)
- $\u22a2\mathbf{FIG}\left(\mathrm{Clo}X\right)$, i.e.,$\mathrm{Clo}X$is a figure.
- (b)
- $\u22a2X\subseteq \mathrm{Clo}X$.
- (c)
- $\u22a2\mathrm{Fig}\left(\mathrm{Clo}X\right)\equiv \mathrm{Clo}X$.
- (d)
- $\u22a2\mathrm{Clo}\left(\mathrm{Clo}X\right)\equiv \mathrm{Clo}X$.
- (e)
- If X is a figure then$\u22a2\mathrm{Clo}X\equiv \mathrm{Clo}\left(\mathrm{Fig}X\right)$.

**Proof.**

**Definition**

**5.**

- (i)
- Interior of a fuzzy set:$${\mathrm{Int}}_{\left(o\alpha \right)\left(o\alpha \right)}:=\lambda {x}_{o\alpha}\lambda {v}_{\alpha}\xb7(\exists {y}_{o\alpha})(\mathbf{FIG}y\wedge (y\subseteq x)\wedge yv).$$
- (ii)
- A fuzzy set Y is dense in X if$$\u22a2(Y\subseteq X)\wedge (X\subseteq \mathrm{Clo}Y).$$

**Theorem**

**2.**

- (a)
- $\u22a2{\mathrm{Int}}_{\left(o\alpha \right)\left(o\alpha \right)}{X}_{o\alpha}\equiv \lambda {v}_{\alpha}\xb7(\exists {u}_{\alpha})(Xu\wedge (\mathrm{Mon}u\subseteq X)\&\left(\mathrm{Mon}u\right)v).$
- (b)
- $\u22a2{\mathrm{Int}}_{\left(o\alpha \right)\left(o\alpha \right)}{X}_{o\alpha}\equiv \lambda {v}_{\alpha}\xb7(\exists {u}_{\alpha})((Xu\wedge (\forall v)(v\approx u\Rightarrow Xv))\&(u\approx v))$.

**Proof.**

**Lemma**

**10.**

**Proof.**

#### 5.3. Rough Fuzzy Sets in FTT

**Definition**

**6.**

- (i)
- Upper approximation of a fuzzy set${x}_{o\alpha}$:$${\mathrm{Up}}_{\left(o\alpha \right)\left(o\alpha \right)}:=\lambda {x}_{o\alpha}\lambda {u}_{\alpha}\xb7(\exists {v}_{\alpha})(xv\&\left(\mathrm{Mon}u\right)v).$$
- (ii)
- Lower approximation of a fuzzy set${x}_{o\alpha}$:$${\mathrm{Lo}}_{\left(o\alpha \right)\left(o\alpha \right)}:=\lambda {x}_{o\alpha}\lambda {u}_{\alpha}\xb7(\forall {v}_{\alpha})(\left(\mathrm{Mon}u\right)v\Rightarrow xv).$$

**Lemma**

**11.**

- (a)
- $\mathrm{Up}X\equiv \lambda {u}_{\alpha}\xb7(\exists {v}_{\alpha})(Xv\&(u\approx v))$.
- (b)
- $\mathrm{Lo}X\equiv \lambda {u}_{\alpha}\xb7(\forall {v}_{\alpha})((u\approx v)\Rightarrow Xv)$.

**Proof.**

**Lemma**

**12.**

- (a)
- $\u22a2\mathbf{FIG}\left(\mathrm{Up}X\right)$, i.e.,$\mathrm{Up}X$is a figure.
- (b)
- $\u22a2\mathbf{FIG}\left(\mathrm{Lo}X\right)$, i.e.,$\mathrm{Lo}X$is a figure.

**Proof.**

**Theorem**

**3.**

**Proof.**

- L.1
- $\u22a2\mathrm{Up}X\equiv u\xb7(\exists v)(Xv\&(u\approx v\left)\right)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{Lemma}\text{}11\left(\mathrm{a}\right)$
- L.2
- $\u22a2\mathrm{Fig}X\equiv \lambda u\xb7(\exists v)(Xv\&(u\approx v\left)\right)\hspace{0.17em}\hspace{0.17em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{Lemma}\text{}4\left(\mathrm{b}\right)$
- L.3
- $\u22a2\mathrm{Up}X\equiv \mathrm{Fig}X\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{L}.1,\text{}\mathrm{L}.2,\text{}\mathrm{Rule}\text{}\left(\mathrm{R}\right)$

**Lemma**

**13.**

- (a)
- $\u22a2\mathrm{Lo}X\subseteq X$and$\u22a2X\subseteq \mathrm{Up}X$.
- (b)
- $\u22a2\mathrm{Lo}\varnothing \equiv \varnothing $and$\u22a2\mathrm{Up}\varnothing \equiv \varnothing $.
- (c)
- $\u22a2\mathrm{Up}(X\cup Y)\equiv (\mathrm{Up}X\cup \mathrm{Up}Y)$.
- (d)
- $\u22a2\mathrm{Lo}(X\cap Y)\equiv (\mathrm{Lo}X\cap \mathrm{Lo}Y)$.
- (e)
- $\u22a2\mathrm{Up}(X\cap Y)\subseteq (\mathrm{Up}X\cap \mathrm{Up}Y)$.
- (f)
- $\u22a2(\mathrm{Lo}X\cup \mathrm{Lo}Y)\subseteq \mathrm{Lo}(X\cup Y)$.
- (g)
- $\u22a2\mathrm{Lo}\left(\mathrm{Lo}X\right)\equiv \mathrm{Lo}X$and$\u22a2\mathrm{Up}\left(\mathrm{Up}X\right)\equiv \mathrm{Up}X$.

**Proof.**

**Theorem**

**4.**

- (a)
- $\u22a2X\equiv \mathrm{Lo}X$,
- (b)
- $\u22a2\mathrm{Lo}X\equiv \mathrm{Up}X.$

**Proof.**

**Definition**

**7.**

**Lemma**

**14.**

- (a)
- If X is a figure, i.e.,$\u22a2\mathbf{FIG}X$, then$\u22a2\mathrm{BN}X\equiv \varnothing $.
- (b)

**Proof.**

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

- (a)
- $\u22a2\mathrm{BN}X\equiv \varnothing $.
- (b)
- $\u22a2\mathrm{Lo}X\equiv \mathrm{Up}X$.
- (c)
- $\u22a2\mathbf{FIG}X$.

**Proof.**

#### 5.4. Model

_{Δ}-algebra and, furthermore,

**Proposition**

**1.**

**Proof.**

- (i)
- Let the interpretation ${\mathcal{M}}_{p}^{R}\left(X\right)=[c1,c2]$ (a set). Then interpretation ${\mathcal{M}}_{p}^{R}\left(\mathrm{Fig}X\right)$ of a figure of X is a fuzzy set depicted in Figure 2.
- (ii)
- Let the interpretation ${\mathcal{M}}_{p}^{R}\left(X\right)$ be$${\mathcal{M}}_{p}^{R}\left(X\right)\left(z\right)=\left\{\begin{array}{cc}1-\frac{{x}_{0}-z}{0.5}\hfill & z\in [{x}_{0}-0.5,{x}_{0}],\hfill \\ 1-\frac{z-{x}_{0}}{0.1}\hfill & z\in [{x}_{0},{x}_{0}+0.1],\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

## 6. Conclusions

## Funding

## Conflicts of Interest

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Novák, V.
Topology in the Alternative Set Theory and Rough Sets via Fuzzy Type Theory. *Mathematics* **2020**, *8*, 432.
https://doi.org/10.3390/math8030432

**AMA Style**

Novák V.
Topology in the Alternative Set Theory and Rough Sets via Fuzzy Type Theory. *Mathematics*. 2020; 8(3):432.
https://doi.org/10.3390/math8030432

**Chicago/Turabian Style**

Novák, Vilém.
2020. "Topology in the Alternative Set Theory and Rough Sets via Fuzzy Type Theory" *Mathematics* 8, no. 3: 432.
https://doi.org/10.3390/math8030432