*2.1. Closeness Space*

We are interested in the notion of closeness on finite sets that, as described below, agrees with [1,2]. To emphasize its general applicability, we express it using the language of classical set theory. We show that it can be also expressed in the languages of fuzzy set theory and graph theory.

**Definition 1** (Closeness)**.** *Let X* = {*xi* | *i* = 1, ... , *L*} *be a set, L* ∈ N*, then* closeness on *X is any symmetric, non-negative function w*: *X*<sup>2</sup> → R*, where symmetry means* ∀*<sup>x</sup>*, *y* ∈ *X* : *<sup>w</sup>*(*<sup>x</sup>*, *y*) = *<sup>w</sup>*(*y*, *x*) *and X*<sup>2</sup> = *X* × *X* = {(*<sup>x</sup>*, *y*)| *x*, *y* ∈ *X*} *is the Cartesian product of X with itself. The pair* (*<sup>X</sup>*, *w*) *is called a* space with closeness *or simply a* closeness space*.*

We provide the following semantic interpretation of closeness: objects that are closer in a certain sense (as explained in the Introduction, in some cases, metric cannot be defined) have a greater closeness value (with respect to natural order on R) than the less-close ones. Non-close objects have a closeness value equal to zero, and, vice versa, if the closeness value of a pair of objects is greater, we consider these objects closer in this sense than a pair of objects with a smaller closeness value. The closeness value hence quantifies a certain quality of mutual alikeness between objects.

**Example 1.** *Let X* = {0, 10, 20, 30, 40, 50} *and for all xi*, *xj* ∈ *X* : *<sup>w</sup>*(*xi*, *xj*) = 1 <sup>1</sup>+|*xi*−*xj*| *, then w is closeness on X and* (*<sup>X</sup>*, *w*) *is a space with closeness.*

**Example 2.** *Let ρ be a metric and* (*X* , *ρ*) *a metric space. Let moreover, X* ⊆ *X and for all xi*, *xj* ∈ *X, <sup>w</sup>*1(*xi*, *xj*) = 1 <sup>1</sup>+*ρ*(*xi*,*xj*) *and <sup>w</sup>*2(*xi*, *xj*) = e<sup>−</sup>*ρ*(*xi*,*xj*) *hold true. Then,* (*<sup>X</sup>*, *<sup>w</sup>*1) *and* (*<sup>X</sup>*, *<sup>w</sup>*2) *are closeness spaces.*

**Definition 2** ((Weakly) Reflexive Closeness)**.** *Let X* = {*xi* | *i* = 1, ... , *L*} *be a set, L* ∈ N*, and w*: *X*<sup>2</sup> → R *be closeness on X. Then, w is* weakly reflexive closeness on *X if there exists a positive real number w, s.t.* ∀*<sup>x</sup>*, *y* ∈ *X* : *<sup>w</sup>*(*<sup>x</sup>*, *y*) ≤ *<sup>w</sup>*(*<sup>x</sup>*, *x*) = *w. If, moreover, w* = 1*, w is called* reflexive closeness on *X. The pair* (*<sup>X</sup>*, *w*) *is then a* space with (weakly) reflexive closeness *or simply a* (weakly) reflexive closeness space*.*

**Remark 1.** *In this paper, we consider only finite closeness domain X, hence there always exists a constant m* = max{*w*(*xi*, *xj*)|(*xi*, *xj*) ∈ *<sup>X</sup>*<sup>2</sup>}*, in case* 1 < *m* < + <sup>∞</sup>*, it would be more appropriate to define strong reflexivity by w* = *m but later, we work with closeness w*: *X*<sup>2</sup> - [0, 1]*.*

Example 2 contains two reflexive closeness-defining functions. Examples of weakly reflexive closenesses derived from an arbitrary metric *ρ* include 1 <sup>2</sup>+*ρ*(*xi*,*xj*) and 12 <sup>e</sup><sup>−</sup>*ρ*(*xi*,*xj*), bothwith

 12 Let (*<sup>X</sup>*, *w*) be a space with closeness where *X* = {*xi* | *i* = 1, ... , *L*} is a set, *L* ∈ N. Let usdenote

$$\forall \mathbf{x}\_{i\prime}, \mathbf{x}\_{j} \in \mathcal{X} \colon w\_{ij} = w(\mathbf{x}\_{i\prime}, \mathbf{x}\_{j}) \,, \tag{1}$$

and consider a matrix that contains the closeness values

$$\mathcal{W} = (w\_{ij})\_{i,j=1}^{L} \in \mathbb{R}^{L \times L},\tag{2}$$

then the matrix *W* is called the *matrix of closeness on X* or simply the *closeness matrix*. Functional values of closeness *w* for the elements of *X*<sup>2</sup> can be arranged into a matrix *W*— the matrix of closeness on *X*.

#### *2.2. Fuzzy Set, Relation and Partition*

*w* =

.

**Definition 3** (Fuzzy Set)**.** *Let X be a non-empty* universe *and K*: *X* → [0, 1] *be a function.* Fuzzy set *K is the set of pairs* {(*<sup>x</sup>*, *<sup>K</sup>*(*x*))| *x* ∈ *<sup>X</sup>*}*. The function K is a* membership function *and its functional value <sup>K</sup>*(*x*) *is called the* membership degree *of the element x to K. Fuzzy set is conventionally identified with its membership function. We call it* fuzzy subset *of X which is denoted by K* ⊂∼*X. We say that the point x* ∈ *X is* covered *by the fuzzy set K if <sup>K</sup>*(*x*) > 0*.*

**Remark 2.** *Note that if K*: *X* → {0, <sup>1</sup>}*, then K is a characteristic function of an ordinary set. The membership function K*: *X* → [0, 1] *is a generalization of the characteristic function of the set K* ⊆ *X given by a function χK* : *X* → {0, <sup>1</sup>}*. If the universe of a (fuzzy) set is clear, we can talk about (fuzzy) set K without referring to it.*

**Definition 4** (Fuzzy Relation)**.** *Let V and X be non-empty sets, then* binary fuzzy relation *r is any fuzzy subset of V* × *X , i.e., r* ⊂∼*V* × *X.*

Hence, the binary fuzzy relation *r* is identified with a membership function *r*: *V* × *X* → [0, 1].

**Definition 5** (Convex Fuzzy Set)**.** *Let X* ⊂ R *and K* ⊂∼ *X. Then K is a* convex fuzzy set*, if* ∀*xc*, *xd*, *xe* ∈ *X*, *xc* < *xd* < *xe* : *<sup>K</sup>*(*xd*) ≥ min{*K*(*xc*), *<sup>K</sup>*(*xe*)}*.*

Following [4], we recall the notion of fuzzy partition and modify it for a discrete case. Consecutively, we set fuzzy transform in Section 2.6.

**Definition 6** (Fuzzy Partition of a Real Interval with Nodes)**.** *Let* [*y*0, *yl*+<sup>1</sup>] *be a real interval and y*1, ... , *yl* ∈ R*, s.t. y*0 < *y*1 < *y*2 < ... < *yl* < *yl*+<sup>1</sup> *and* 0 < *l* < +<sup>∞</sup>*. Then the* fuzzy partition *of* [*y*0, *yl*+<sup>1</sup>] *is given by a set of fuzzy sets* {*<sup>A</sup>*1,..., *Al*}*, if:*


*Elements of the set* {*<sup>A</sup>*1, ... , *Al*} *are called* basic functions (fuzzy partition units)*. Based on condition 3., A*1, ... , *Al are associated with points y*1, ... , *yl, respectively. These points are called* nodes *of the fuzzy partition.*

In Definition 6, conditions 3–5 ensure that each basic function is a convex fuzzy set. Moreover, condition 3 ensures that each basic function covers at least one point. This might not hold in the case of a discrete universe which motivated the following definition.

**Definition 7** (Sufficient Density with respect to Fuzzy Partition)**.** *Let X be a set, s.t. X* ⊂ [*y*0, *yl*+<sup>1</sup>] ⊂ R*, and* 0 < *l* < +<sup>∞</sup>*. Let* P = {*<sup>A</sup>*1, ... , *Al*} *be a fuzzy partition of the interval* [*y*0, *yl*+<sup>1</sup>]*. Then X is* sufficiently dense *with respect to* P*, if* ∀*i* = 1, ... , *l* ∃*x* ∈ *X* : *Ai*(*x*) > 0*.*

Definition 7 states that a set is sufficiently dense with respect to a given fuzzy partition if every basic function covers at least one point of that set. This notion allows us to define the fuzzy partition of a discrete set.

**Definition 8** (Fuzzy Partition of a Discrete Subset of a Real Interval with Nodes)**.** *Let X* = {*<sup>x</sup>*1,..., *xL*} *be a set, s.t. X* ⊂ [*y*0, *yl*+<sup>1</sup>] ⊂ R*, and* 0 < *l* ≤ *L* < +<sup>∞</sup>*. Let* P = {*A*1,..., *<sup>A</sup>l*} *be a fuzzy partition of the interval* [*y*0, *yl*+<sup>1</sup>] *with nodes in Y* = {*y*1, ... , *yl*} ⊆ *X and X be sufficiently dense with respect to* P*. Then* P = {*<sup>A</sup>*1, ... , *Al*} *is the* fuzzy partition of *X* with nodes in *Y , if* ∀*i* = 1, . . . , *l* : *Ai* = *<sup>A</sup>i*|*X, i.e., if each basic function is replaced by its restriction on X.*

**Remark 3.** *Definition 8 can be extended in the converse direction: let X* = {*<sup>x</sup>*1, ... , *xL*} *be a set, s.t. X* ⊂ [*y*0, *yl*+<sup>1</sup>] ⊂ R*, Y* = {*y*1, ... , *yl*} ⊆ *X and* 0 < *l* ≤ *L* < +<sup>∞</sup>*. Let* P = {*<sup>A</sup>*1, ... , *Al*}*, s.t.* ∀*i* = 1, ... , *l* : *Ai* ⊂∼ *X* & *Ai*(*yi*) > 0*. Then* P *will be also called the fuzzy partition of X with nodes in Y, if there exists a fuzzy partition* P = {*A*1, ... , *<sup>A</sup>l*} *with nodes in Y, s.t. X is sufficiently dense with respect to* P *and for each i* = 1, ... , *l, Ai is a continuous extension of Ai on* [*y*0, *yl*+<sup>1</sup>]*, i.e., Ai* ⊂∼[*y*0, *yl*+<sup>1</sup>]*, Ai is continuous and Ai* = *<sup>A</sup>i*|*X.*

Below, we show how the space with closeness can be represented by a fuzzy partitioned space and by a weighted graph-structured space that, satisfying certain conditions, are examples of closeness spaces.

#### *2.3. Closeness as Fuzzy Relation and Closeness Given by a Fuzzy Partition*

Using the language of fuzzy set theory, we can say that any symmetric fuzzy relation *w* ⊂∼*X*2, where symmetry means ∀*<sup>x</sup>*, *y* ∈ *X* : *<sup>w</sup>*(*<sup>x</sup>*, *y*) = *<sup>w</sup>*(*y*, *<sup>x</sup>*), is closeness on *X*.

Note that in the context of fuzzy relations, it holds that for weakly reflexive closeness, we have *w* ≤ 1. For *w* = 1, we obtain a special case called reflexive closeness on *X*.

**Lemma 1** (Closeness Given by a Fuzzy Partition – 1)**.** *Let* {*<sup>A</sup>*1, ... , *Al*} *be a fuzzy partition associated with nodes y*1, ... , *yl of the set* {*<sup>x</sup>*1, ... , *xL*} ⊂ [*y*0, *yl*+<sup>1</sup>]*, s.t. y*0,..., *yl*+1, *x*1,..., *xL* ∈ R*,* 0 < *l* = *L* < +∞ *and y*0 < *y*1 = *x*1 < *y*2 = *x*2 < ··· < *yl* = *xL* < *yl*+1*. For all i*, *j* = 1, ... , *l, let it hold that*

$$A\_i(y\_j) = A\_j(y\_i) \, , \tag{3}$$

*then the function w*: {*<sup>x</sup>*1, ... , *xL*}<sup>2</sup> → R *given by* ∀*i*, *j* = 1, ... , *L*: *<sup>w</sup>*(*xi*, *xj*) = *Ai*(*xj*) *is closeness on* {*<sup>x</sup>*1,..., *xL*}*.*

**Proof.** The function *w* is obviously real, bivariate and non-negative. Using the property (3), we have ∀*i*, *j* = 1, ... , *L*: *<sup>w</sup>*(*xi*, *xj*) = *Ai*(*xj*) = *Ai*(*yj*) = *Aj*(*yi*) = *Aj*(*xi*) = *<sup>w</sup>*(*xj*, *xi*), hence *w* is also symmetric.

Lemma 1 shows that a fuzzy partition of a subset of a real interval, s.t. its every point is a node and satisfying condition (3), determines closeness on this set.

#### *2.4. Closeness as Weighted Adjacency of Graph Vertices*

In this subsection, we show that closeness can be expressed using the language of graph theory. We do not recall the basic concepts of this theory (details can be found, e.g., in [20]). Generally, closeness between any two objects *xi* and *xj* in *X* is described by the values (weights) *<sup>w</sup>*(*xi*, *xj*) of a real, non-negative, symmetric bivariate function *w*.

Let *<sup>G</sup>*(*<sup>X</sup>*, *E*, *W*) be a weighted graph where the set of vertices corresponds to objects in *X*, the set of edges *E* ⊆ *X*<sup>2</sup> and weights of the edges are given by the adjacency matrix *W* and in accordance with (2). The objects are non-close if and only if the corresponding vertices are not connected by a direct edge, i.e., there is a fictional edge with zero weight between them. We denote this disconnectedness or non-closeness by ∼ which is a classical (meaning non-fuzzy) symmetric binary relation on *X*, i.e., ∼⊆ *X*2:

$$\mathbf{x}\_{i}\nleftarrow{\swarrow} \mathbf{x}\_{j}\Leftrightarrow w(\mathbf{x}\_{i},\mathbf{x}\_{j}) = \mathbf{0}.\tag{4}$$

In the other words, the value of closeness determines the existence of an edge.

By defining the values of closeness {*w*(*xi*, *xj*)| *xi*, *xj* ∈ *<sup>X</sup>*}, i.e., by defining the values of entries of the closeness matrix, we simultaneously define the values of entries of the adjacency matrix *W* which uniquely determines a weighted graph *<sup>G</sup>*(*<sup>X</sup>*, *E*, *W*) by (1), (2) and (4). That is why the notion of closeness can be also established from the perspective of weighted graphs: let *X* = {*xi* | *i* = 1, ... , *L*} be a set of vertices, *L* ∈ N, then, following the convention given by (4) as *E* ⊆ *X*2, any evaluation of the set of all edges *X*<sup>2</sup> described by a symmetric, non-negative function *w*: *X*<sup>2</sup> → R is closeness on *X*.

Below, we join the theory of closeness with a certain type of fuzzy partition to describe the space this paper deals with.
