*2.5. Initial Assumptions*

Let us have a finite set of points on R:

$$X = \{ \mathbf{x}\_i \mid i = 1, \dots, L \}, \tag{5}$$

and its non-empty subset

$$\mathcal{Y} \subseteq X, \quad 0 < |\mathcal{Y}| = l \le L = |X| < +\infty.$$

Let {*At* | *t* ∈ *Y*} be a fuzzy partition of *X* with nodes in *Y* (in the sense of Remark 3), s.t. with each node *t* ∈ *Y*, we associate a basic function *At* : *X* → [0, 1]. Let, moreover, (3) be satisfied, i.e., ∀*t*,*<sup>s</sup>* ∈ *Y*: *At*(*s*) = *As*(*t*), and let all points in *X* be covered by at least one basic function, i.e.,

$$\bigcup\_{t \in Y} \{ \mathbf{x} \in X \mid A\_t(\mathbf{x}) > 0 \} = X. \tag{6}$$

Let us define *node degrees* by

$$\forall t \in \mathcal{Y} \colon d\_{tt} = \sum\_{\mathbf{x} \in \mathcal{X}} A\_t(\mathbf{x}) \,. \tag{7}$$

 .

Following condition 3 in Definition 6, we know that *At*(*t*) > 0, *t* ∈ *Y*, which implies that each node degree *dtt* ∈ (0, <sup>∞</sup>). Moreover, based on Definition 7, it means that the set *Y* is sufficiently dense with respect to the fuzzy partition {*At* | *t* ∈ *<sup>Y</sup>*}; hence, the same holds for the universe *X*.

It is worth noting that instead of working with the closeness on the whole universe *X*, given by a function on *X*2, we work only with its restriction on *Y* × *X* (the values of closeness on (*X* \ *Y*) × *X* are undefined, as explained in the Introduction). In the other words, closeness *w*: *X*<sup>2</sup> - R is a partial function (partial function *f* : *A* - *B* maps each element of a set *A* to at most one element of the set *B*, it is a functional binary relation with carrier set being a subset of *A* × *B*) given by

$$\forall \mathbf{x}\_{i}, \mathbf{x}\_{j} \in X \colon w(\mathbf{x}\_{i}, \mathbf{x}\_{j}) = \begin{cases} A\_{t}(\mathbf{x}\_{j}) & t = \mathbf{x}\_{i} \in \mathcal{Y} \\ \text{undefined} & \text{otherwise} \end{cases}$$

This description justifies the selection of nodes within the universe (Semantic description of nodes is also explained in the Introduction).

Even though we will still call the total function *w*: *Y* × *X* → R (total function *f* : *A* → *B* maps each element of a set *A* to exactly one element of the set *B*, it is a classical function) *closeness on X* and denote the corresponding closeness space as (*<sup>X</sup>*, *<sup>w</sup>*).

**Lemma 2** (Closeness Given by a Fuzzy Partition—2)**.** *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*, y*0 < *y*1 < *y*2 < ... < *yl* < *yl*+1*, y*0 < *x*1 < *x*2 < ... < *xL* < *yl*+1*,* {*y*1, ... , *yl*} ⊆ {*<sup>x</sup>*1,..., *xL*}*,* 0 < *l* ≤ *L* < +∞ *and let condition* (3) *hold. Let us denote*

$$\forall i = 1, \ldots, L: \kappa(i) = \begin{cases} j & \mathbf{x}\_i = y\_j \\ l+1 & \nexists y \in \{y\_1, \ldots, y\_l\}: \mathbf{x}\_i = y\_i \end{cases}$$

*(note that in case xi* = *yj, we have yj* = *<sup>y</sup>κ*(*i*)*). Then the function w*: {*<sup>x</sup>*1, ... , *xL*}<sup>2</sup> - R *given by <sup>w</sup>*(*xi*, *xj*) = *<sup>A</sup>κ*(*i*)(*xj*)*, where κ*(*i*) ≤ *l, 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>κ</sup>*(*i*), *κ*(*j*) ≤ *l* : *<sup>w</sup>*(*xi*, *xj*) = *<sup>A</sup>κ*(*i*)(*xj*) = *<sup>A</sup>κ*(*i*)(*<sup>y</sup>κ*(*j*)) = *<sup>A</sup>κ*(*j*)(*<sup>y</sup>κ*(*i*)) = *<sup>A</sup>κ*(*j*)(*xi*) = *<sup>w</sup>*(*xj*, *xi*), hence *w* is also symmetric.

Lemma 2 shows that a fuzzy partition of a subset of a real interval, s.t. it contains all nodes and satisfies condition (3), determines closeness on this set. We will follow Lemma 2, and hence closeness is given by:

$$w(t, \mathbf{x}) = A\_l(\mathbf{x}), \quad \text{where } t \in \mathcal{Y}, \mathbf{x} \in \mathcal{X}. \tag{8}$$

By that, the closeness space (*<sup>X</sup>*, *w*) is specified. The values of closeness *w*: *Y* × *X* → R are inserted in the closeness matrix *W* ∈ R*l*×*<sup>L</sup>* and given by

$$\forall t \in \mathcal{Y}, \mathbf{x} \in X \colon \mathcal{W}(t, \mathbf{x}) = w(t, \mathbf{x}) \,. \tag{9}$$

It means that by fixing a set of basic functions, we determine values of the closenessdescribing function *w*: *Y* × *X* → R.

It follows from (4), (6) and (8) that for every point *x* of the universe *X* there exists a node *t* ∈ *Y*, s.t. *<sup>w</sup>*(*<sup>t</sup>*, *x*) > 0. It means that every point is connected with at least one node.

Sections 2.1 and 2.2 covered the basic theory of closeness and the theory of fuzzy partitions, respectively. In the following Section 2.6, we recall the theory of fuzzy transform that we need to describe a fundamental property (the F-transform identity) of the space with closeness that will be used in Section 3 to describe the preimage problem.
